\documentstyle[12pt,dina4]{article} 
 
\advance\voffset by -2cm                                           
                                                                                
\clubpenalty=10000                                                              
\widowpenalty=10000                                                             
                                                                                
\newenvironment{lemma}{\begingroup\samepage\begin{lemmma}\ }{\end{lemmma}%      
     \endgroup}                                                                 
\newtheorem{lemmma}{Lemma}[section]                                             
\newenvironment{proof}{\begin{prooof}\rm\ \nopagebreak}{\end{prooof}}           
\newcommand{\proofend}{\qquad\ifmmode\Box\else$\Box$\fi}                        
\newtheorem{prooof}{Proof}                                                      
\renewcommand{\theprooof}{}   % makes shure that prooof doesn't get numbers       
\newenvironment{Figure}{\begin{figure}\vspace{1\baselineskip}}%                 
                       {\vspace{1\baselineskip}\end{figure}}                    
\newlength{\figspace}         % space between figures in a single               
\setlength{\figspace}{30pt}   % Figure environment                              
                                                                                
\newcommand{\var}[1]{{\it #1\/}}     % use it for names of variables            
\newcommand{\emph}[1]{{\em #1\/}}    % use it for emphazided text               
                                     % (This notion sticks to the               
                                     % applicative style of markup.)            
\renewcommand{\O}{{\rm O}}           % O-notation, also for math mode           
\newcommand{\T}{{\cal T}}            % the set T in math mode                   
\newcommand{\TreeTeX}{Tree\TeX}                                                 
\newcommand{\fig}[1]{Figure~\ref{#1}}                                           
\let\p\par                                                                      
                                                                                
\input TreeTeX                                    
\Treestyle{\vdist{20pt}\minsep{16pt}}                                                        
\dummyhalfcenterdim@n=2pt                                                       
                                                                                
\def\Node(#1,#2){\put(#1,#2){\circle*{4}}}                                      
\def\Edge(#1,#2,#3,#4,#5){\put(#1,#2){\line(#3,#4){#5}}}                        
                                                                                
\def\enode{\node{\external\type{dot}}}                                          
\def\inode{\node{\type{dot}}}                                                   
                                                                                
\def\e{\node{\external\type{dot}}}                                              
\def\i{\node{\type{dot}}}                                                       
\def\il{\node{\type{dot}\leftonly}}                                             
\def\ir{\node{\type{dot}\rightonly}}                                            
                                                                                
\newcommand{\stack}[3]{%                                                        
     \vtop{\settowidth{\hsize}{#1}%                                             
     \setlength{\leftskip}{0pt plus 1fill}%                                     
     \setlength{\baselineskip}{#2}#3}}                                          
                                                                                
\let\multic\multicolumn                                                         
                                                                                
\newlength{\hd} % hidden digit                                                  
\setbox0\hbox{1}                                                                
\settowidth{\hd}{\usebox{0}}                                                    
\newcommand{\ds}{\hspace{\hd}} % digit space                                    
                                                                                
\newcommand{\ccol}[1]{\multicolumn{1}{c}{#1}}                                   
                                                                                
\hyphenation{post-or-der sym-bol Karls-ruhe bool-ean}                                    
                                                                                
\begin{document}                                                                
                                                                                
\bibliographystyle{plain}                                                       
                                                                                
\title{Drawing Trees Nicely with \TeX\thanks{This work was supported by         
     a Natural Sciences and Engineering Research Council of Canada              
     Grant~A-5692 and a Deutsche Forschungsgemeinschaft Grant~Sto167/1-1.
     It was started during the first author's stay with              
     the Data Structuring Group in Waterloo.}}                        
\author{Anne Br\"uggemann-Klein\thanks{Institut f\"ur Informatik,     
     Universit\"at Freiburg, Rheinstr.~10--12, 7800~Freiburg,
     West~Germany}\ \and Derick Wood\thanks{Data          
     Structuring Group, Department of Computer Science, University of           
     Waterloo, Waterloo, Ontario, N2L~3G1, Canada}}                             
\maketitle                                                                      
                                                                                
\begin{abstract}                                                                
                                                                                
Various algorithms have been proposed for the difficult problem of              
producing aesthetically pleasing drawings of trees, see~%                       
\cite{TidierTrees,TidyTrees} but                                                
implementations only exist as ``special purpose software'',                     
designed for special environments. Therefore,                                   
many users resort to the                                                        
drawing facilities available on most personal computers, but the                
figures obtained in this way still look ``hand-drawn''; their quality is        
inferior to the quality of the surrounding text that can be realized by         
today's high quality text processing systems.                                   
                                                                                
In this paper we present an entirely new solution that                          
integrates a tree drawing algorithm into one of the best text                   
processing systems available. More precisely, we present a \TeX{} macro package 
\TreeTeX{} that produces a drawing of a tree from a purely logical              
description. Our approach has three advantages. First, labels           
for nodes can be handled in a reasonable way. On the one hand, the tree         
drawing algorithm can compute the widths of the labels and take                 
them into account for the positioning of the nodes; on the other hand,          
all the textual parts of the document can be treated uniformly. Second,         
\TreeTeX{} can be trivially ported to any site running \TeX{}. Finally,         
modularity in the description of a tree and \TeX{}'s macro capabilities         
allow for libraries of subtrees and tree classes.                               
                                                                                
In addition, we have implemented an option that produces                        
drawings which make the                                                         
\emph{structure} of the trees more obvious to the human eye,                    
even though they may not be as aesthetically pleasing.                          
                                                                                
\end{abstract}                                                                  
                                                                                
\section{Aesthetical criteria for drawing trees}                                
                                                                                
One of the most commonly used data structures in computer science is the tree.  
As many people are using trees in their research or just as illustration        
tools, they are usually struggling with the problem of                          
\emph{drawing} trees. We are concerned primarily with ordered                   
trees in the sense of~\cite{ACP}, especially binary and unary-binary            
trees. A binary tree is a finite set of nodes which either                      
is empty, or consists of a root and two disjoint binary trees called            
the left and right subtrees of the root. A unary-binary tree is                 
a finite set of nodes which either is empty, or consists of a root and          
two disjoint unary-binary trees, or consists of a root and one                  
nonempty unary-binary tree. An extended binary tree is a binary tree            
in which each node has either two nonempty subtrees or two                      
empty subtrees.                                                                 
                                                                                
For these trees there                                                           
are some basic agreements on how they should be drawn, reflecting               
the top-down and left-right ordering of nodes in a tree;                        
see \cite{TidierTrees} and \cite{TidyTrees}.                                    
                                                                                
\begin{enumerate}                                                               
\item[1.] Trees impose a distance on the nodes; no node                         
          should be closer to the root than any of its                          
          ancestors.                                                            
\item[2.] Nodes of a tree at the same height should lie on a straight           
          line, and the straight lines defining the levels should be            
          parallel.                                                             
\item[3.] The relative order of nodes on any level should be the same           
          as in the level order traversal of the tree.                          
\end{enumerate}                                                                 
                                                                                
These axioms guarantee that trees are drawn as planar graphs: edges do          
not intersect except at nodes. Two further axioms improve the aesthetical       
appearance of trees:                                                            
                                                                                
\begin{enumerate}                                                               
\item[4.] In a unary-binary tree, each left child should be positioned          
          to the left of its parent, each                                       
          right child to the right of its parent, and each unary child          
          should be positioned below its parent.                                
\item[5.] A parent should be centered over its children.                        
\end{enumerate}                                                                 
                                                                                
An additional axiom deals with the problem of tree drawings becoming too wide   
and therefore exceeding the physical limit of the output medium:                
                                                                                
\begin{enumerate}                                                               
\item[6.] Tree drawings should occupy as little width as possible without       
          violating the other axioms.                                           
\end{enumerate}                                                                 
                                                                                
In \cite{TidyTrees}, Wetherell and Shannon introduce two algorithms for         
tree drawings, the first of which fulfills axioms~1--5, and the second          
1--6. However, as Reingold and Tilford in \cite{TidierTrees}                    
point out, there is a lack of symmetry in the algorithms of                     
Wetherell and Shannon which may lead to unpleasant results.
Therefore, Reingold and Tilford introduce a new structured        
axiom:                                                                          
                                                                                
\begin{enumerate}                                                               
\item[7.] A subtree of a given tree should be                                   
          drawn the same way regardless of where it occurs in the given tree.   
\end{enumerate}                                                                 
                                                                                
Axiom~7 allows the same tree to be drawn differently when it occurs as          
a subtree in different trees.                                                   
Reingold and Tilford give an algorithm which fulfills axioms~1--5               
and~7. Although                                                                 
this algorithm doesn't fulfill axiom~6,      
the aesthetical improvements are well worth the additional space.
\fig{algorithms} illustrates the benefits of axiom~7, and \fig{narrowtrees}
shows that the algorithm of Reingold and Tilford violates axiom~6.
 
\begin{Figure}                                                                  
\centering
\leavevmode\noindent                                                                      
\begin{Tree}
\enode
\enode\enode\inode\enode\enode\inode\inode\inode
\node{\external\type{dot}\rght{\unskip\hskip2\mins@p\hskip2\dotw@dth}}
\enode\enode\inode\enode\enode\inode\inode\inode
\inode                                                                    
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\begin{Tree}                                                               
\enode
\enode\enode\inode\enode\enode\inode\inode\inode
\enode
\enode\enode\inode\enode\enode\inode\inode\inode
\inode
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\                                                               
\caption{The left tree is drawn by the algorithm of Wetherell and Shannon,
and the tidier right one is drawn by the algorithm of Reingold and Tilford.}             
\label{algorithms}                                                                  
                                                                                
\vspace{\figspace}                                                              
\centering
\leavevmode\noindent
\begin{Tree}
\enode\enode\enode\enode\enode\enode\enode\enode\enode
\enode\inode\inode\inode                                                                      
\enode\inode\inode\inode                                                                      
\enode\inode\inode\inode                                                                      
\enode\inode\inode\inode 
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\begin{Tree}
\enode\enode\enode\enode\enode\enode\enode\enode
\node{\external\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
\enode\inode\inode\inode 
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\
\caption{The left tree is drawn by the algorithm of Reingold and Tildford, but
the right tree shows that narrower drawings fulfilling all aesthetic axioms
are possible.}
\label{narrowtrees}                                                                  
\end{Figure}                                                                    
                                                                                
                                                                                
\section{The algorithm of Reingold and Tilford}                                 
                                                                                
The algorithm of Reingold and Tilford (hereafter called ``the RT~algorithm'')   
takes a modular approach to the                                                 
positioning of nodes: The relative positions of the nodes in a subtree          
are calculated independently from the rest of the tree. After the               
relative positions of two subtrees have been calculated, they can be            
joined as siblings in a larger tree by placing them as close                    
together as possible and centering the parent node above them.                  
Incidentally, the modularity principle is the reason that the                   
algorithm fails to fulfill axiom~6; see~\cite{Complexity}.                      
Two sibling subtrees are placed as close together as possible,                           
during a postorder traversal, as follows. At each node \var{T},                 
imagine that its two subtrees have been drawn and cut out of paper along        
their contours. Then, starting with the two subtrees superimposed at their      
roots, move them apart until a minimal agreed upon distance                     
between the trees is obtained at each level. This can be done gradually:        
Initially, their roots are separated by some agreed upon minimum                
distance. Then, at the next lower level,                                        
they are pushed                                                                 
apart until the minimum separation is established there.                        
This process is continued at successively lower levels until the                
bottom of the shorter subtree is reached. At some levels no movement may be     
necessary; but at no level are the two subtrees moved closer                    
together. When the process is complete, the position of the                     
subtrees is fixed relative to their parent, which is centered over them.        
Assured that the subtrees will never be placed closer together,                 
the postorder traversal is continued.                                           
                                                                                
A nontrivial implementation of                                                  
this algorithm has been obtained by Reingold and Tilford that runs              
in time $\O(N)$, where $N$ is the number of                                     
nodes of the tree to be drawn.                                                  
Their crucial idea is to keep track of the contour of the subtrees              
by special pointers, called threads, such that whenever                         
two subtrees are joined, only the                                               
top part of the trees down to the lowest level of the                           
smaller tree need to be taken into account.                              
                                                                                
The RT algorithm is given in \cite{TidierTrees}.
The nodes are positioned on a fixed grid and are             
considered to have zero width. No labelling is provided. The algorithm only     
draws binary trees, but is easily extendable to multiway trees.                 
                                                                                
\section{Improving human perception of trees}                                   
 
It is common understanding in book design that aesthetics and readability
don't necessarily coincide, and---as Lamport (\cite{LaTeX}) puts it---%
books are meant to be read, not to be hung on walls. Therefore, readability is
more important than aesthetics.
 
When it comes to tree drawings, readability means that the structure of
a tree must be easily recognizable. This criterion is not always met
by the RT~algorithm. As an example, there are trees whose structure is very
different, the only common thing being the fact that they have the same number
of nodes at each level. The RT~algorithm might assign identical positions to
these nodes making it very hard to perceive the different structures.
Hence, we have modified the RT~algorithm such that additional white space
is inserted between subtrees of 
\emph{significant} nodes. Here a binary node 
is called significant if the minimum distance
between its two subtrees is taken \emph{below} their root level.
Setting the amount of additional white space to zero retains the original RT~%
placement. The effect of having nonzero additional white space between 
the subtrees of significant
nodes is illustrated in \fig{addspace} .
 
Another feature we have added to the RT~algorithms is the possibility to draw
an unextended binary tree with the same placement of nodes as its
associated extended version. We define the \emph{associated extended version}
of a binary tree to be the binary tree obtained by replacing each empty subtree
having a nonempty sibling with a subtree consisting of one node. This feature
also makes the structure of a tree more prominent; see \fig{extended}.
 
\begin{Figure}
\centering
\leavevmode\noindent                                                                      
\begin{Tree} 
\e\il\e\e\i\i\il % the left subtree
\e\ir\il % the right subtree  
\i
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\begin{Tree}
\e\il\il\il % the left subtree
\e\e\i\e\i\il % the right subtree
\i
\end{Tree}                                                               
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\adds@p10pt
\begin{Tree}
\e\il\e\e\i\node{\type{dot}\lft{$\longrightarrow$}}\il % the left subtree
\e\ir\il % the right subtree  
\node{\type{dot}\lft{$\longrightarrow$}}
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\begin{Tree}
\e\il\il\il % the left subtree
\e\e\i\e\i\il % the right subtree
\node{\type{dot}\lft{$\longrightarrow$}}
\end{Tree}                                                               
\hskip\leftdist\box\TeXTree\hskip\rightdist\
\adds@p0pt
 
\caption{The first two trees get the same placement of their nodes
by the RT~algorithm, although the structure of the two trees is very different.
The alternative drawings highlight the structure of the trees by adding
additional white space between the subtrees of 
($\longrightarrow$) significant nodes.}
\label{addspace}                                                                                
\end{Figure} 
 
\begin{Figure}
\centering
\leavevmode\noindent
\begin{Tree}
\e\e\i\il\e\e\i\i
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\begin{Tree}
\e\e\i\e\i\e\ir\i
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\extended
\begin{Tree}
\e\e\i\il\e\e\i\i
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
\begin{Tree}
\e\e\i\e\i\e\ir\i
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\\
\noextended
\begin{Tree}
\e\e\i\e\i\e\e\i\i
\end{Tree}
\hskip\leftdist\box\TeXTree\hskip\rightdist\  
\caption{In the first two drawings, the RT~algorithm assigns the same placement 
to the nodes of two trees although their structure is very different. The modified
RT~algorithms highlights the structure of the trees by optionally
drawing them like their extended
counterpart, which is given in the second row.}
\label{extended}
\end{Figure}
 
 
\section{Trees in a document preparation environment}                           
                                                                                
Drawings of trees usually don't come alone, but are included in some text       
which is itself typeset by a text processing system. Therefore, a typical      
scenario is a pipe of three stages. First comes the tree drawing             
program which calculates the positioning of the nodes of the tree to            
be drawn and outputs a description of the tree drawing in                       
some graphics language; next comes a graphics system which transforms this      
description into an intermediate language which can be interpreted by the output
device; and finally comes the                                                   
text processing system which integrates the output of the                       
graphics system into the text.                                 
                                                                                
This scenario loses its linear structure once nodes have to be labelled, since
the labelling influences the positioning of the nodes. Labels usually occur
inside, to the left of, to the right of, or beneath nodes (the latter only for
external nodes), and their extensions certainly should be taken into account
by the  tree drawing algorithm. But the labels have to be typeset first
in order to determine their extensions,
preferably by the typesetting program that
is used for the regular text, because this method makes for the uniformity in the textual
parts of the document and provides the author with the full power of the
text processing system for composing the labels. Hence, a more complex
communication scheme than a simple pipe is required.
 
Although a system of two processes running simultaneously might be the most
elegant solution, we wanted a system that is easily portable to
a large range of hardware at our sites 
including personal computers with single process
operating systems.                                                           
Therefore, we thought of using a text processing system                              
having programming facilities powerful enough to program a tree drawing algorithm                      
and graphics facilities powerful enough                                                             
to draw a tree. One text processing system                  
rendering outstanding typographic quality and good enough programming           
facilities is \TeX, developed by Knuth at Stanford University;                  
see~\cite{TeXbook}.                                                             
The \TeX{} system includes the following programming facilities:                
                                                                                
\begin{enumerate}                                                               
\item[1.] datatypes:\\                                                          
     integers~(256), dimensions\footnote{The term \emph{dimension} is used      
     in \TeX\ to describe physical measurements of typographical objects,       
     like the length of a word.}~(512), boxes~(256), tokenlists~(256), boolean  
     variables~(unrestricted)                                                   
\item[2.] elementary statements:\\                                              
     $a:=\rm const$, $a:=b$ (all types);\\                                      
     $a:=a+b$, $a:=a*b$, $a:=a/b$ (integers and dimensions);\\                  
     horizontal and vertical nesting of boxes                                   
\item[3.] control constructs:\\                                                 
     if-then-else statements testing relations between integers,                
     dimensions, boxes, or boolean variables                                    
\item[4.] modularization constructs:\\                                          
     macros with up to 9~parameters (can be viewed as procedures without        
     the concept of local variables).                                               
\end{enumerate}                                                                 
                                                                                
Although the programming                                                      
facilities of \TeX{} hardly exceed the abilities of a Turing machine,
they are sufficient to
handle relatively small programs. How about the graphics facilities?   
Although \TeX{} has no built-in graphics facilities, it                         
allows the placement of characters in arbitrary positions on                    
the page. Therefore, complex pictures can be synthesized from elementary        
picture elements treated as characters. Lamport has included such               
a picture drawing environment in his macro package \LaTeX, using                
quarter circles of different sizes and line segments (with and without          
arrow heads) of different slopes as basic elements; see~\cite{LaTeX}.           
These elements are sufficient for drawing trees.                                
                                                                                
This survey of \TeX's capabilities implies that \TeX{} may be a suitable        
text processing system to implement a tree drawing algorithm directly.          
We are basing our algorithm on the RT~algorithm, because this algorithm               
gives the aesthetically most pleasing results. In the first version
presented here, we         
restrict ourselves to unary-binary trees, although our method is                
applicable to arbitrary multiway trees. But in order to take advantage          
of the text processing environment, we expand the algorithm to allow            
labelled nodes.                                                                 
                                                                                
In contrast to previous tree drawing programs, we feel no necessity to          
position the nodes of a tree on a fixed grid. While this may be                 
reasonable for a plotter with a coarse resolution, it is certainly not          
necessary for \TeX, a system that is capable of handling                        
arbitrary dimensions                                                            
and produces device \emph{independent} output.                            
 
                                                                                
\section{A representation method for \TeX{}trees}                               
                                                                                
The first problem to be solved in implementing our tree drawing algorithm       
is how to choose a good internal representation                                 
for trees. A straightforward adaptation                                         
of the implementation by Reingold and Tilford requires, for each node,          
at least the following fields:                                                  
                                                                                
\begin{enumerate}                                                               
\item two pointers to the children of the node                                  
\item two dimensions for the offset to the left and the right child (these      
      may be different once there are labels of different widths to the         
      left and right of the nodes)                                              
\item two dimensions for the $x$- and $y$-coordinates of the final              
      position of the nodes                                                     
\item three or four labels                                                      
\item one token to store the geometric shape (circle, square, framed text etc.)
      of the node.                       
\end{enumerate}                                                                 
                                                                                
Because these data are used very frequently in calculations, they should be     
stored in registers (that's what variables are called in \TeX),
rather than being recomputed, in order to obtain           
reasonably fast performance. This gives a total of $10N$ registers for          
a tree with $N$ nodes, which would exceed                                                          
\TeX's limited supply of registers. Therefore, we present a
modified algorithm hand-tailored to the abilities of \TeX{}.
We start with the following observation.                   
Suppose a unary-binary tree is constructed bottom-up, in a postorder            
traversal. This is done by iterating the following three steps in               
an order determined by the tree to be constructed.                              
                                                                                
\begin{enumerate}                                                               
\item Create a new subtree consisting of one external node.                     
\item Create a new subtree by appending the two subtrees created last           
      to a new binary node; see \fig{Construct}.                               
\item Create a new subtree by appending the subtree created last as a left,     
      right, or unary subtree of a new node; see \fig{Construct}.              
\end{enumerate}                                                                 
                                                                                
(A pointer to) each subtree that has been                                       
created in steps 1--3 is pushed onto a stack, and                               
steps 2 and 3 remove two trees or one, respectively,                            
from the stack before the push                                                  
operation is carried out. Finally, the tree to be constructed will              
be the remaining tree on the 
stack.                                             
                                                                                
\begin{Figure}                                                                  
\centering                                                                      
\begin{Tree}                                                                    
\treesymbol{\lvls{2}}%                                                          
\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}                    
$+$                                                                             
\treesymbol{\lvls{2}}%                                                          
\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}\quad               
$\Longrightarrow$\quad                                                          
\treesymbol{\lvls{2}}%                                                          
\treesymbol{\lvls{2}}%                                                          
\node{\type{dot}}%                                                              
\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
\end{Tree}
 
\vskip\baselineskip                                                                      
 
\begin{Tree}                                                                    
\treesymbol{\lvls{2}}%                                                          
\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}\quad               
$\Longrightarrow$\quad                                                          
\treesymbol{\lvls{2}}%                                                          
\node{\leftonly\type{dot}}%                                                     
\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
\quad or\quad                                                                   
\treesymbol{\lvls{2}}%                                                          
\node{\unary\type{dot}}%                                                        
\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
\quad or\quad                                                                   
\treesymbol{\lvls{2}}%                                                          
\node{\rightonly\type{dot}}%                                                    
\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
\end{Tree}                                                                      
                                                                     
\caption{Construction steps 2 and 3}                                                   
\label{Construct}                                                              
\end{Figure}                                                                    
                                                                                
This tree traversal is performed twice in the RT~algorithm.
During the first pass,                            
at each execution of step 2 or step 3, the relative positions of the         
subtree(s) and of the new node are computed. 
A closer examination of the RT~algorithm reveals that information about the
subtree's coordinates is not needed during this pass; the contour information
alone would be sufficient. Complete information is only needed in the second
traversal, when the tree is actually drawn. Here a special feature of 
\TeX{} comes in that allows us to save registers.
Unlike Pascal, \TeX{} provides the capability of
storing a drawing in a single box register that can be positioned freely in
later drawings. This means that in our implementation the two passes
of the original RT~algorithm can be intertwined into a single pass,
storing for each subtree on the stack its contour and its drawing.
Although the latter is a complex object, it takes only one of
\TeX's precious registers.
 
                                                                              
\section{The internal representation}                                           
                                                                                
Given a tree, the corresponding \TeX{}tree is a box containing                  
the ``drawing'' of the tree, together with some additional                      
information about the contour of the tree.                                                              
The reference point of a \TeX{}tree-box is always in the root of the            
tree. The height, depth, and width of the box of a \TeX{}tree are               
of no importance in this context.                                               
                                                                                
The additional information about the contour of the tree is stored in some      
registers for numbers and dimensions and                                        
is needed in order to put subtrees together to form a larger tree.              
\var{loff} is an array of dimensions which contains for each                    
level of the tree the horizontal offset between the                             
left end of the                                                                 
leftmost node at the current level and the                                      
left end of the leftmost node at                                                
the next level.                                                                 
\var{lmoff} holds the horizontal offset between the root                        
and the leftmost node of the whole tree. \var{lboff} holds the                  
horizontal offset between the root and the leftmost node at                     
the bottom level of the tree.                                                   
Finally, \var{ltop} holds the distance between the reference point              
of the tree and the leftmost end of the root.                                   
The same is true for                                                            
\var{roff}, \var{rmoff}, \var{rboff}, and \var{rtop}; just replace              
``left'' by ``right''. Finally,                                                 
\var{height} holds the height of the tree, and \var{type} holds the             
geometric shape of the root of the tree. \fig{TeXtree} shows an example \TeX{}tree,
i.e. a tree drawing and the corresponding additional information.
 
\begin{Figure}
\centering                                                                      
\begin{Tree}                                                                    
\e\ir\ir\e                                                                      
  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}% 
  \i % A                                                                        
\end{Tree}                                                                      
\leavevmode                                                                     
\stack{-10pt}{\vd@st}{%                                                
     -10pt\\10pt\\10pt\\\var{loff}}%                              
\hspace{1em}%                                                                   
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
\hspace{1em}%                                                                   
\stack{-10pt}{\vd@st}{%                                                         
     15pt\\5pt\\-10pt\\\var{roff}}% 
 
\vskip\baselineskip\raggedright
height:~3, type:~dot, ltop:~2pt, rtop:~2pt, lmoff:~-10pt, rmoff:~20pt, lboff:~10pt,
rboff:~10pt.
 
\caption{A \TeX{}tree consists of the drawing of the tree and the
additional information. The width of the dots is 4pt, the minimal separation between
adjacent nodes is 16pt, making for a distance of 20pt center to center.
The length of the small rule labelling one of the nodes is 5pt. The column left (right)
of the tree drawing is the array \var{loff} (\var{roff}),
describing the left (right) contour of the tree. At each level,
the dimension given is the horizontal
offset between the border at the current and at the next level. The offset between
the left border of the root node and the leftmost node at level~1 is -10pt,
the offset between the right border of the root node and the rightmost node at
level~1 is 15pt, etc.}                               
\label{TeXtree}
\end{Figure}
                                                                                
Given two \TeX{}trees \var{A} and \var{B},                                      
how can a new \TeX{}tree \var{C} be built that                                  
consists of a new root and has \var{A} and \var{B} as subtrees?                 
An example is given in \fig{AddInfo}.                        
                                                                                
\begin{Figure}                                                                  
\centering                                                                      
\begin{Tree}                                                                    
\e\ir\ir\e                                                                      
  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}% 
  \i % A                                                                        
\end{Tree}                                                                      
\leavevmode                                                                     
A: \stack{-10pt}{\vd@st}{%                                                      
     -10pt\\10pt\\10pt\\\ \\\var{loff}(\var{A})}%                              
\hspace{1em}%                                                                   
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
\hspace{1em}%                                                                   
\stack{-10pt}{\vd@st}{%                                                         
     15pt\\5pt\\-10pt\\\ \\\var{roff}(\var{A})}%                               
\qquad                                                                          
\begin{Tree}                                                                    
\e\il\e\i\il\il\ir % B                                                          
\end{Tree}                                                                      
\leavevmode                                                                     
B: \stack{-10pt}{\vd@st}{%                                                      
     10pt\\-10pt\\-10pt\\-10pt\\-10pt\\\ \\\var{loff}(\var{B})}%               
\hspace{1em}%                                                                   
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
\hspace{1em}%                                                                   
\stack{-10pt}{\vd@st}{%                                                         
     10pt\\-10pt\\-10pt\\10pt\\-30pt\\\ \\\var{roff}(\var{B})}%                
\\[\figspace]                                                                        
\begin{Tree}                                                                    
\e\ir\ir\e                                                                      
  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}% 
  \i % A                                                                        
\e\il\e\i\il\il\ir % B                                                          
\i % C                                                                          
\end{Tree}                                                                      
\leavevmode                                                                     
C: \stack{-10pt}{\vd@st}{%                                                      
     -20\\-10pt\\%                                                              
     \makebox[0pt][r]{\var{loff}(\var{A})$\smash{\left\{\vrule height\vd@st     
          depth\vd@st width0pt\right.}$ }%                                      
     10pt\\10pt\\%                                                              
     \makebox[0pt][r]{$\longrightarrow$ }%                                      
     10pt\\%                                                                    
     \makebox[0pt][r]{\raisebox{-.5\vd@st}{\var{loff}(\var{B})$\smash           
          {\left\{\vrule height.5\vd@st                                         
          depth.5\vd@st width0pt\right.}$ }}%                                   
     \makebox[0pt][r]{-}10pt\\\ \\\var{loff}(\var{C})}%                        
\hspace{1em}%                                                                   
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
\hspace{1em}%                                                                   
\stack{-10pt}{\vd@st}{%                                                         
     20pt\\10pt\\-10pt\\-10pt%                                                  
     \makebox[0pt][l]{\raisebox{-.5\vd@st}{                                     
          $\smash{\left\}\vrule height2.5\vd@st                                 
          depth2.5\vd@st width0pt\right.}$\var{roff}(\var{B})}}%                
     \\10pt\\-30pt\\\ \\\var{roff}(\var{C})}%                                  
                                                                                
\vspace{\figspace}                                                              
\centering                                                                      
\begin{tabular}{|l|r|r|r|}                                                      
\hline                                                                          
&\multic{1}{c|}{\var{A}}&\multic{1}{c|}{\var{B}}&\multic{1}{c|}{\var{C}}\\      
\hline                                                                          
height&\multic{1}{c|}{3}&  \multic{1}{c|}{5}&  \multic{1}{c|}{6}\\              
type&  \multic{1}{c|}{dot}&\multic{1}{c|}{dot}&\multic{1}{c|}{dot}\\            
ltop&  2pt&                2pt&                2pt\\                            
rtop&  2pt&                2pt&                2pt\\                            
lmoff& -10pt&              -30pt&              -30pt\\                          
rmoff& 20pt&               10pt&               30pt\\                           
lboff& 10pt&               -30pt&              -10pt\\                          
rboff& 10pt&               -30pt&              -10pt\\                          
\hline                                                                          
\end{tabular}\qquad                                                             
\begin{tabular}{|c|r|r|}                                                        
\hline                                                                          
\multic{1}{|c|}{level}&\multic{1}{c|}{\var{totsep}}&                            
                      \multic{1}{c|}{\var{currsep}}\\                           
\hline                                                                          
0&20pt&0/16pt\\                                                                 
1&25pt&11/16\\                                                                
2&40pt&1/16pt\\                                                                
3&40pt&16pt\\                                                                
\hline                                                                          
\end{tabular}                                                                   
\caption{The \TeX{}trees \var{A} and~\var{B} are combined to form the
larger \TeX{}\-tree~\var{C}. The small table gives the 
history of computation for \var{totsep} and \var{currsep}.}                  
\label{AddInfo}                                                                 
\end{Figure}                                                                    
 
                                                                                
First we determine which tree is higher; this is                                
\var{B} in the example.                                                         
Then we have to compute the minimal distance                                    
between the roots of \var{A} and \var{B}, such that at all levels               
of the trees there is free space of at least \var{minsep} between               
the trees when they are drawn side by side.                                     
For this purpose we keep track of two values, \var{totsep} and                  
\var{currsep}. The variables \var{totsep} and \var{currsep}                     
hold the total distance between the roots and the distance                      
between the rightmost node of \var{A} and the leftmost node                     
of \var{B} at the current level. In order to calculate                          
\var{totsep} and \var{currsep}, we start at level 0 and                         
visit each level of the trees until we reach the bottom level                   
of the smaller tree; this is \var{A} in our example.                            
                                                                                
At level 0, the distance between the roots of \var{A} and \var{B}               
should be at least \var{minsep}. Therefore, we set                              
$\var{totsep}:=\var{minsep} + \var{rtop}(\var{A})                               
+ \var{ltop}(\var{B})$ and $\var{currsep}:=\var{minsep}$.                       
Using $\var{roff}(\var{A})$ and $\var{loff}(\var{B})$, we can                   
proceed to calculate \var{currsep} for the next level.                          
If $\var{currsep} < \var{minsep}$, we have to increase \var{totsep} by          
the difference and update \var{currsep}. This process is                        
iterated until we reach the lowest level of \var{A}.                            
Then \var{totsep} holds the final distance between the                          
nodes of \var{A} and \var{B}, as calculated by the RT~algorithm. 
If the root of \var{C} is a significant node, then the additional space ,
which is 0pt by default, is added to \var{totsep}.               
However, the approach of synthesizing                                           
drawings from simple graphics characters allows only a finite                   
number of orientations for the tree edges; therefore, \var{totsep}              
must be increased slightly to fit the next orientation                          
available.                                                                      
                                                                                
Now we are ready to construct the box of \TeX{}tree~\var{C}.                    
Simply put \var{A} and~\var{B} side by side, with the reference                      
points \var{totsep}~units apart, insert a new node                              
above them, and connect the parent and children by edges.                       
                                                                                
Next, we update the additional information                                      
for \var{C}. This can be done by using the additional information               
for \var{A} and~\var{B}.                                                        
Note that most components of $\var{roff}(\var{C})$ and                          
$\var{lroff}(\var{C})$ are the same as in the higher tree, which                
is \var{B} in our case.                                                         
So, if we can avoid moving this information around, we only have                
to access $\var{height}(\var{A}) + \var{const}$ many counters in                
order to update the additional information for \var{C}.                         
This implies that we can apply the same argument as                             
in~\cite{TidierTrees}, which gives                                              
us a running time of $\O(N)$ for drawing a tree with N nodes.                   
                                                                                
Therefore, we must carefully design the storage allocation for                  
the additional information of \TeX{}trees in order to fulfill the               
following requirements:                                                         
If a new tree is built from                                                     
two subtrees, the additional information of the new tree should                 
share storage with its larger subtree.                                          
Organizational overhead, that is,                                               
pointers which keep track of the locations of different parts of additional     
information, must be avoided.                                                   
This means that all the additional information                                  
for one \TeX{}tree should be stored in a row of consecutive dimension registers           
such that only one pointer granting access to the first element                        
in this row is needed.                                                          
On the other hand, each parent                                                  
tree is higher and therefore needs more storage than its subtrees.              
So we must ensure that there is always enough space in the row                  
for more information.                                                           
                                                                                
The obvious way to fulfill these requirements is to use a stack and to           
allow only the topmost \TeX{}trees of this stack to be                          
combined into a larger tree at any time.                   
This leads to the following register allocation: A subsequent number of         
box registers contains the treeboxes of the subtrees in the stack. A            
subsequent number of token registers contains the type information for the      
nodes of the subtrees in the stack. For each subtree in the stack,              
a subsequent number of dimension registers contains the contour                 
information of the subtree. The ordering of these groups of dimension           
registers reflects the ordering of the subtrees in the                          
stack. Finally, a subsequent number of counter registers contains               
the height and the address of the first dimension register for                  
each subtree in the stack. Four address counters store the addresses            
of the last treebox, type information, height, and address of contour           
information. A sketch of the register organization for a stack of \TeX{}trees
is provided in \fig{Registers}.
 
\begin{Figure}                                                                  
Dimension registers\\                                                           
\var{lmoff}(1) \var{rmoff}(1) \var{lboff}(1) \var{rboff}(1) \var{ltop}(1)       
               \var{rtop}(1)\\                                                  
\var{loff}($h_1$) \var{roff}($h_1$) \dots\ \var{loff}(1) \var{roff}(1)\\        
\dots\\                                                                         
\var{lmoff}($n$) \var{rmoff}($n$) \var{lboff}($n$) \var{rboff}($n$)             
                 \var{ltop}($n$) \var{rtop}($n$)\\                              
\var{loff}($h_n$) \var{roff}($h_n$) \dots\ \var{loff}(1) \var{roff}(1)\\        
\ \\                                                                            
Counter registers\\                                                             
\var{lasttreebox} \var{lasttreeheight} \var{lasttreeinfo} \var{lasttreetype}\\  
\var{treeheight}(1) \var{diminfo}(1) \dots\ \var{treeheight}($n$)               
                    \var{diminfo}($n$)\\                                        
\ \\                                                                            
Box registers\\                                                                 
\var{treebox}(1) \dots\ \var{treebox}($n$)\\                                    
\ \\                                                                            
Token registers\\                                                               
\var{type}(1) \dots\ \var{type}($n$)
 
\caption{\var{lasttreebox}, \var{lasttreeheight}, \var{lasttreeinfo},                    
\var{lasttreetype} contain pointers to \var{treebox}($n$)                        
\var{treeheight}($n$), \var{lmoff}($n$), \var{type}($n$),                       
\var{diminfo}($i$) contains a pointer to                                        
\var{lmoff}($i$). Unused dimension registers are                                
allowed between the dimension registers of subsequent trees. The counter
registers \var{lasttreebox},\ldots,\var{diminfo}($n$) serve as a directory
mechanism to access the \TeX{}trees on the stack.}                   
\label{Registers}                                                               
\end{Figure}                                                                    
                                                                   
                                                                                
When a new node is pushed onto the stack, the treebox, type information,        
height, address of contour information, and contour information are             
stored in the next free registers of the appropriate type, and the              
four address counters are updated accordingly.                                  
                                                                                
When a new tree is formed from the topmost subtrees on the stack,               
the treebox, type information, height, and address of contour information       
of the new tree are sorted in the registers formerly used by the bottommost     
subtree that has occured in the construction step, and the four address registers are       
updated accordingly. This means that these informations for the subtrees        
are no longer accessible. The contour information of the new subtree            
is stored in the same registers as the contour information of the larger        
subtree used in the construction, apart from the left and right offset          
of the root to the left and right child, which are stored in the                
following dimension registers. That means that gaps can occur                   
between the contour information of subsequent subtrees in the                    
stack, namely when the right subtree, which is on a higher position on the      
stack, is higher than the left one. In order to avoid these                     
gaps, the user can specify an option \verb.\lefttop. when entering a            
binary node, which makes the topmost tree in the stack the                      
left subtree of the node.                                                       
                                                                                
This stack concept also has consequences for the design of the user interface   
that is discussed in Section~\ref{Interface}.                                   
                                                                                
\section{Space cost analysis}                                                   
                                                                                
Suppose we want to draw a unary-binary tree $T$ of height $h$ having            
$N$ nodes\footnote{The height $h$ and the number of nodes $N$ refer to the
drawing of the tree. $N$ is the number of circles, squares etc.~actually
drawn, and $h$ is the number of levels in the drawing minus 1.}.
According to our internal representation,                            
for each subtree in the stack we need                                           
                                                                                
\begin{enumerate}                                                               
\item one box register to store the box of the \TeX{}tree.                       
\item one token register to store the type of the root of the subtree.           
\item $2h^\prime+6$ dimension registers to store the additional 
      information, where $h^\prime$ is the height of the                        
      subtree.                                                                   
\item three counter registers to store the register numbers of the             
      box register, the token register, and the first dimension register above.         
\end{enumerate}                                                                 
                                                                                
The following lemma relates to $h$ and $N$ the number
of subtrees of $T$ which are on the      
stack simultaneously and their heights.                          
                                                                                
\begin{lemma}                                                                   
\begin{enumerate}                                                               
\item At any time, there are at most $h+1$ subtrees of $T$ on the               
      stack.                                                                    
\item For each set $\T$ of subtrees of $T$ which are on the stack               
      simultaneously we have                                                    
      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)                             
        \le\min(N,{(h+1)(h+2)\over2}).$$                                      
\end{enumerate}                                                                 
\end{lemma}                                                                     
                                                                                
\begin{proof}                                                                   
\begin{enumerate}                                                               
\item By induction on $h$.\label{stackdepth}                                           
\item The trees in $\T$ are pairwise disjoint, and each tree of                 
      height $h^\prime$ has at least $h^\prime+1$ nodes. This implies           
      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)                             
        \le N.$$                                                                
      The second part is shown by induction on $h$.                           
      The basis $h=0$ is clear.                                                 
      Assume the assumption holds for all trees of height less than             
      $h$. If $\T$                                                              
      contains only subtrees of either the left or the right subtree            
      of $T$, we have                                                           
      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)\le                          
      {h(h+1)\over2}\le{(h+1)(h+2)\over2}.$$                                    
      Otherwise, $\T$ contains the left or the right subtree $T_s$ of           
      $T$. Then all elements of $\T-\{T_s\}$ belong to the other                
      subtree. This implies                                                     
      \begin{eqnarray*}                                                         
      \sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)&\le&                          
      {\rm ht}(T_s)+1                                                           
      +\sum_{T^\prime\in \T-\{T_s\}}({\rm ht}(T^\prime)+1)\\                    
      &\le& h+{h(h+1)\over2}\le{(h+1)(h+2)\over2}.\proofend                       
      \end{eqnarray*}                                                           
\end{enumerate}                                                                 
\end{proof}                                                                     
                                                                                
Therefore, our implementation uses at most $9h+2\min(N,(h+1)(h+2)/2)$          
registers. In order to compare this with the                                    
$10N$ registers used in the straightforward implementation,                     
an estimation of the average height of a tree with $N$ nodes is                 
needed. Several results, depending on the type of trees and of the                       
randomization model, are cited in \fig{Stat}, which 
compares the number of registers used in a straightforward           
implementation with the average number of registers used in our                 
implementation. This table shows clearly the advantage of our                  
implementation. 
 
\begin{Figure}
\centering                                                                  
\begin{tabular}{|c|c|c|c|c|}                                                    
\hline                                                                          
&registers&\multicolumn{3}{c|}{average registers}\\                             
\cline{3-5}                                                                     
nodes&(straight-&extended&unary-binary&binary\\                           
&forward)&binary trees&trees&                                            
     search trees\\
&&($\sqrt{\pi n}$) \cite{AverageHeight}&
  ($\sqrt{3\pi n}$) ~\cite{BinaryTrees}&
  ($4.311\log n$) \cite{BinarySearchTrees}\\                                                             
\hline                                                                          
\ds8&    \ds80& \ds61.12& \ds94.15& \ds51.04\\                                  
\ds9&    \ds90& \ds65.86&   100.89& \ds55.02\\                                  
  10&      100& \ds70.44&   107.37& \ds58.80\\                                  
  11&      110& \ds74.91&   113.64& \ds62.41\\                                  
  12&      120& \ds79.26&   119.71& \ds65.87\\                                  
  20&      200&   111.34&   163.56& \ds90.48\\                                  
  30&      300&   147.37&   211.33&   117.31\\                                  
  40&      400&   180.89&   254.75&   132.58\\                                  
  50&      500&   212.80&   295.37&   143.54\\                                  
\hline                                                                          
\end{tabular}
                                                                   
\caption{The numbers of registers used by a straightforward implementation
(second column) and by our modified implementation (third to fifth column)
of the RT~algorithm are
given for different types of trees and randomization models.
The formula in parentheses indicates the average height of the respective class
of trees, as depending on the number of nodes.}                                                        
\label{Stat}                                                                    
\end{Figure}                                                                    
                                                                
                                                                                
\section{The user interface}\label{Interface}                                   
                                                                                
\subsection{General design considerations}                                      
                                                                                
The user interface of \TreeTeX{} has been designed in the spirit of             
the thorough separation of the logical description of document components       
and their layout; see~\cite{DocumentFormatting,GML}. This concept               
ensures both uniformity and flexibility of document layout and frees            
authors from layout problems which have nothing to do with the                  
substance of their work. For some powerful implementations and projects         
see \cite{Tables,Karlsruhe,LaTeX,Grif,Scribe}.                                  
                                                                                
In this context, the description of a tree is given in a purely                 
logical form, and layout variations are defined by a separate style             
command which is valid for all trees of a document.                             
                                                                                
A second design principle is to provide defaults for all specifications,        
thereby allowing the user to omit many definitions                              
if the defaults match what he or she wants.                                                        
                                                                                
The node descriptions of a tree must be entered in postorder.                   
This fits the internal representation                                      
of \TeX{}trees best. Although this is a natural method of describing a               
tree, a user might prefer more flexible description methods.                     
However, note that instances of well defined tree classes can be described      
easily by \TeX{} macros. In section~\ref{ExampleClasses}. we give examples of macros                             
for complete binary trees and Fibonacci trees.                                  
                                                                                
\TreeTeX{} uses the picture making macros of \LaTeX. If \TreeTeX{} is used with 
any other macro package or format, the picture macros of                        
\LaTeX{} are included automatically.                                            
                                                                                
\subsection{The description of a tree}                                          
                                                                                
The description of a tree is started by the command \verb.\beginTree.           
and closed by \verb.\endTree. (or \verb.\begin{Tree}. and                       
\verb.\end{Tree}. in \LaTeX). The description can be                  
started in any mode; it defines a box and two dimensions. The                   
box is stored in the box register \verb.\TeXTree. and contains the              
drawing of the tree. The box has zero height and width, and its depth           
is the height of the drawing. The reference point is in the                     
center of the node of the tree. The dimensions are stored in the                
registers \verb.\leftdist. and \verb.\rightdist. and describe                   
the distance between the reference point and the left and                       
right margin of the drawing. These data can be used to position the             
drawing of the tree.                                                            
                                                                                
Note that the \TreeTeX{} macros don't contribute anything to the current        
page but only store their results in the registers                              
\verb.\TeXTree., \verb.\leftdist., and \verb.\rightdist.. It is the             
user's job to put the drawing onto the page, using the                          
commands \verb.\copy. or \verb.\box. (or \verb.\usebox. in \LaTeX).             
                                                                                
Each matching pair of \verb.\beginTree. and \verb.\endTree. must                
contain the description for only \emph{one} tree.                               
Descriptions of trees cannot be nested and                                      
new registers cannot be allocated inside                                        
a matching pair of \verb.\beginTree. and \verb.\endTree..                       
                                                                                
As already stated, each tree description defines the nodes of the tree in                        
postorder, that is, a tree description is a particular sequence of node         
descriptions.                                                                   
                                                                                
A node description, in turn, consists of the macro \verb.\node.,                
followed by a list of node options, included in braces. The list                
of node options may be empty. The node options describe the labels,             
the geometric shape (type), and the outdegree of the node. Default values are   
provided for all options which are not explicitly specified.                    
The following node options are available:                                       
                                                                                
\begin{enumerate}                                                               
\item[1.] \verb.\lft{<label>}., \verb.\rght{<label>}.,                          
     \verb.\cntr{<label>}.,\\\verb.\bnth{<label>}.:\\                           
     These options describe the labels which are put to the left of, to the     
     right of, in the                                                           
     center of, or beneath the node (the latter only makes sense for            
     external nodes). The arguments of these macros are processed in            
     internal horizontal mode (LR-mode in \LaTeX), but can consist of           
     arbitrary nested boxes for more sophisticated labels. For each of          
     these options, the default is an empty label.                              
                                                                                
\item[2.] \verb.\external., \verb.\unary.,                                      
     \verb.\leftonly., \verb.\rightonly.:\\                                     
     These options describe the outdegree                                       
     of the node.                                   
     The default is binary (no outdegree option is specified).                  
                                                                                
\item[3.] \verb.\type{<type>}.:\\                                               
     This option describes the type or geometric shape of the node.             
     \verb.<type>. can have the values \verb.square.,                         
     \verb.dot., \verb.text., or \verb.frame.. 
     The default value is \verb.circle. (no type is specified). A node of type
     \verb.square. has a fixed width, while a node of type \verb.frame. has its
     width determined by the center label. A node of type \verb.text. has no frame
     around its center label. The center label can have arbitrary width.
 
\item[3.] \verb.\leftthick., \verb.\rightthick.:
     These options change the thickness of the left or right outgoing edge of
     a binary node. Defaults are thin edges (neither option is specified).      
                                                                                
\item[4.] \verb.\lefttop.:\\                                                    
     The node option \verb.\lefttop. in a binary node makes the                 
     last entered subtree the left child of the node (the right child is the    
     default). This option helps to cut down on the number of dimension registers  
     used during the construction of a tree. As a rule of thumb,                
     this option is recommended when the left subtree has a smaller             
     height than the right subtree, that is,                                    
     in this case the right subtree should                                      
     be entered before the left one and their parent should be assigned the option 
     \verb.\lefttop..                                                           
\end{enumerate}                                                                 
                                                                                
\subsection{Macros for classes of trees}
\label{ExampleClasses}                                        
                                                                                
Tree descriptions can be produced by macros. This is especially useful          
for trees which belong to a larger class of trees and which can be specified    
by some simple parameters. A small library of such                              
macros is provided in the file \verb!TreeClasses.tex!.                          
                                                                                
\begin{enumerate}                                                               
\item[1.] \verb.\treesymbol{<node options>}.:\\                                 
     This macro produces a triangular tree symbol which can be included in      
     a tree description instead of an external node. Labels for these           
     tree symbols are described as for ordinary nodes. In addition, the         
     options \verb.\lvls{<number>}. and \verb.\slnt{<number>}.                  
     are provided. \verb.\lvls. defines the number of levels in the             
     tree over which the triangle extends, and \verb.\slnt. gives               
     the slant of the sides of the triangle, ranging from 1~(minimal)           
     to 24~(maximal). On the other hand,                                        
     \verb.\treesymbol. does not expand to a tree description, because          
     a tree symbol cannot be built from subtrees, and, on the other hand,       
     it is not a node, because it is allowed to extend over several tree        
     levels and therefore has a longer contour than an ordinary node.           
                                                                                
\item[2.] \verb.\binary{<bin specification>}.:\\                                
     This macro truly expands to a tree description. It produces                
     a complete binary tree, that is, an extended binary tree,                  
     where, for a given $h$, all external nodes appear at level $h$             
     or $h-1$, and all external nodes at level $h$ lie left of those at         
     level $h-1$. \verb.<bin specification>. consists of the                    
     following options:                                                         
     \verb.\no{<number>}. defines the number of internal nodes,                 
     with \verb.<number>. greater than 0, and                                   
     \verb.\squareleaves. produces leaves of type                               
     \verb.square.. Defaults are \verb.\no{1}. and                              
     leaves of type \verb.circle..                                              
                                                                                
\item[3.] \verb.fibonacci{<fib specification>}.:\\                              
     This macro produces a Fibonacci tree.              
     \verb.<fib specification>. allows for the three options                    
     \verb.\hght{<number>}., \verb.\unarynodes.,                                 
     and \verb.\squareleaves..                                                  
     Normally, a Fibonacci tree of height $h+2$ is a binary tree                
     with Fibonacci trees of height $h$ and $h+1$ as left and                   
     right subtrees. The option \verb.\unarynodes. means that the               
     Fibonacci tree is augmented by unary nodes such that each                  
     two subtree siblings have the same height. These are examples
     of what has been called brother-trees in the literature; 
     see~\cite{Brother}. Defaults are                    
     \verb.\hght{0}., the unaugmented version of a Fibonacci tree,              
     and external nodes of type \verb.circle..                                  
\end{enumerate}                                                                 
                                                                                
\subsection{Style options for trees}
\sloppy                                            
The \TreeTeX{} package includes a style command                           
\verb.\Treestyle{<style option>}., where \verb.<style option>.                   
contains all the parameter settings the user might want                         
to change.                                                                      
Normally, the command \verb.\Treestyle. appears only once at the beginning      
of the document and the style options are valid for all trees of the            
document.      
 
\fussy                                                                                
The changes in the style options are global. A \verb.\Treestyle. command        
changes only the specified style options; non-specified options retain          
the last specified value or the default value, respectively. The following            
style options are available:                                                    
                                                                                
\begin{enumerate}                                                               
\item[1.] \verb.\treefonts{<font options>}.:\\                                  
     \sloppy                                                                    
     \verb.\treefonts. is invoked by \verb.\beginTree., and it simply executes  
     whatever is specified in \verb.<font options>.. Defaults are               
     \verb.\treefonts{\tenrm}. (or \verb.\treefonts{\normalsize\rm}. in         
     \LaTeX).                                                                   
                                                                                
\fussy                                                                          
\item[2.] \verb.\nodesize{<size>}.:\\                                           
     \verb.\nodesize. defines the size of the nodes. \verb.<size>. is a         
     dimension and specifies the diameter of circle nodes. The                  
     width of square nodes is adjusted accordingly to be slightly               
     smaller than the diameter of circle nodes in order to                      
     balance their appearance. Furthermore,                                     
     \verb.\nodesize. adjusts the amount of space by which the                  
     baseline of the labels is placed beneath the center of the node.           
     The default value of \verb.\nodesize. suits the default of                 
     \verb.\treefonts. (taking into account the size option                     
     of \LaTeX's document style).                                               
                                                                                
\item[3.] \verb.\vdist{<dimen>}., \verb.\minsep{<dimen>}.,                      
     \verb.\addsep{<dimen>}.:\\                                                 
     \sloppy                                                                    
     \verb.vdist. specifies the vertical distance between two                   
     subsequent levels of the tree. Default is \verb.\vdist{60pt}..             
     \verb.\minsep. specifies the minimal horizontal distance between two       
     adjacent nodes. Default is \verb.\minsep{20pt}..                           
     \verb.\addsep. specifies the additional amount of horizontal space         
     by which two subtree siblings are pushed apart farther than                
     calculated by the RT~algorithm,                                            
     if the level at which they are closest is beneath                          
     their root level. Default is \verb.\addsep{0pt}.                           
                                                                                
\fussy                                                                          
\item[4.] \verb.\extended., \verb.\nonextended.:\\                              
     With the option \verb.\extended. in effect, the nodes of a binary          
     tree are placed in exactly the same way as they would be in the            
     associated extended version of the tree (the missing nodes are             
     assumed to have no labels). The default is \verb.\nonextended.,            
     that is the usual layout.                                                  
\end{enumerate}                                                                 
                                                                                
Some examples of tree descriptions                                              
are given in the next figures.                
A detailed description of the                                                   
\TreeTeX{} macros is given in~\cite{TreeTeX}.
                                   
\Treestyle{\vdist{60pt}}                                                        
\dummyhalfcenterdim@n=10pt                                                      
                                                                                
\begin{Figure}
\centering
\begin{Tree}
\node{\external\bnth{first}\cntr{1}\lft{Beeton}}                         
\node{\external\cntr{3}\rght{Kellermann}}                                       
\node{\cntr{2}\lft{Carnes}}                                                     
\node{\external\cntr{6}\lft{Plass}}                                             
\node{\external\bnth{last}\cntr{8}\rght{Tobin}}                          
\node{\cntr{7}\rght{Spivak}}                                                    
\node{\leftonly\cntr{5}\rght{Lamport}}                                          
\node{\cntr{4}\rght{Knuth}}                                                     
\end{Tree}
                                                                        
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
 
\begin{verbatim}
\begin{Tree}
\node{\external\bnth{first}\cntr{1}\lft{Beeton}}                         
\node{\external\cntr{3}\rght{Kellermann}}                                       
\node{\cntr{2}\lft{Carnes}}                                                     
\node{\external\cntr{6}\lft{Plass}}                                             
\node{\external\bnth{last}\cntr{8}\rght{Tobin}}                          
\node{\cntr{7}\rght{Spivak}}                                                    
\node{\leftonly\cntr{5}\rght{Lamport}}                                          
\node{\cntr{4}\rght{Knuth}}                                                     
\end{Tree}
                                                                        
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
\end{verbatim} 
 
\caption{This is an example of a tree that includes labels.}                                         
\end{Figure}
 
\begin{Figure}
\centering
\begin{Tree}
\node{\external\type{frame}\bnth{first}\cntr{Beeton}}                         
\node{\external\type{frame}\cntr{Kellermann}}                                       
\node{\type{frame}\cntr{Carnes}}                                                     
\node{\external\type{frame}\cntr{Plass}}                                             
\node{\external\type{frame}\bnth{last}\cntr{Tobin}}                          
\node{\type{frame}\cntr{Spivak}}                                                    
\node{\leftonly\type{frame}\cntr{Lamport}}                                          
\node{\type{frame}\cntr{Knuth}}                                                     
\end{Tree}
                                                                        
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
 
\begin{verbatim}
\begin{Tree}
\node{\external\type{frame}\bnth{first}\cntr{Beeton}}                         
\node{\external\type{frame}\cntr{Kellermann}}                                       
\node{\type{frame}\cntr{Carnes}}                                                     
\node{\external\type{frame}\cntr{Plass}}                                             
\node{\external\type{frame}\bnth{last}\cntr{Tobin}}                          
\node{\type{frame}\cntr{Spivak}}                                                    
\node{\leftonly\type{frame}\cntr{Lamport}}                                          
\node{\type{frame}\cntr{Knuth}}                                                     
\end{Tree}
                                                                        
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
\end{verbatim}
\caption{This is an example of a tree with framed center labels.} 
\end{Figure}
 
\begin{Figure}                                                                  
\centering                                                                      
\begin{Tree}                                                                    
\binary{\no{6}\squareleaves}                                                    
\end{Tree}
                                                                      
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
                                                                     
\begin{verbatim}                                                                
\begin{Tree}                                                                    
\binary{\no{6}\squareleaves}                                                    
\end{Tree}
                                                                      
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
\end{verbatim}                                                                  
\caption{This is an example of a complete binary tree.}                                     
\end{Figure} 
 
                                                                   
\begin{Figure}                                                                  
\centering                                                                      
\begin{Tree}                                                                    
\fibonacci{\hght{4}\unarynodes\squareleaves}                                    
\end{Tree}
 
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
                                                                      
\begin{verbatim}                                                                
\begin{Tree}                                                                    
\fibonacci{\hght{4}\unarynodes\squareleaves}                                    
\end{Tree}
\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
\end{verbatim}
                                                                  
\caption{This is an example of a Fibonacci tree.}                                           
\end{Figure}                                                                    
 
                                                                                
\clearpage                                                                      
\bibliography{trees}                                                            
\end{document}                                                                  
                                                                                
                                                                                
