                                                                                
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%%% Complete binary trees                                                  %%%  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
                                                                                
% The macro \b@nary{<number>} expands to the description of a complete          
% binary tree with <number> many internal nodes, where each level is filled with
% the maximal number of internal nodes, and the last level of internal nodes    
% is filled from left to right.                                                 
                                                                                
\newcount\b@nno % number of nodes                                               
\newcount\b@nlv % number of complete levels                                     
\newcount\b@ndl % number of nodes on incomplete level                           
                                                                                
\def\ld(#1,#2,#3){% #1, #2, and #3 must be counter registers.                   
                  % #1 is the input, #1 must be >= 1.                           
                  % \ld makes the following assignments:                        
                  % #2:=|_log_2(#1)_|, #3:=2^#2.                                
                  % The contents of #1 is destroyed during the computation.     
     #2=0 #3=1                                                                  
       \loop\ifnum #1>\@ne\relax                                                
            \divide #1 by\tw@ % this is integer division                        
            \advance #2 by\@ne                                                  
            \multiply #3 by\tw@                                                 
     \repeat}                                                                   
                                                                                
\def\b@nary#1{% draws a complete binary tree with #1 internal nodes,            
           % a complete binary tree with N internal nodes has                   
           % lv:=|_log_2(N+1)_| many                                            
           % complete level of binary nodes and dl:=N-2^{lv}+1 many internal    
           % nodes on an incomplete level.                                      
     \b@nno=#1\relax\advance\b@nno by \@ne                                      
     \ld(\b@nno,\b@nlv,\b@ndl)%                                                 
     \b@ndl=-\b@ndl\advance\b@ndl by #1\advance\b@ndl by\@ne                    
     \b@n}                                                                      
                                                                                
\def\b@n{%                                                                      
     \ifnum\b@nlv>\@ne                                                          
           \advance\b@nlv by-\@ne                                               
           \b@n                                                                 
           \b@n                                                                 
           \advance\b@nlv by\@ne                                                
           \node{}                                                              
      \else\ifnum\b@ndl>\@ne                                                    
                 \advance\b@ndl by-\tw@                                         
                 \node{\le@f\external}\node{\le@f\external}\node{}%             
                 \node{\le@f\external}\node{\le@f\external}\node{}%             
                 \node{}%                                                       
            \else\ifnum\b@ndl=\@ne                                              
                       \advance\b@ndl by-\@ne                                   
                       \node{\le@f\external}\node{\le@f\external}\node{}%       
                       \node{\le@f\external}%                                   
                       \node{}%                                                 
                  \else\node{\le@f\external}\node{\le@f\external}\node{}%       
                    \fi                                                         
              \fi                                                               
        \fi}                                                                    
                                                                                
\def\circleleaves{\def\le@f{\type{circle}}}                                     
\def\squareleaves{\def\le@f{\type{square}}}                                     
                                                                                
\newcount\no@                                                                   
\def\no#1{\no@=#1\relax}                                                        
                                                                                
\def\binary#1{%                                                                 
     \no{1}\circleleaves                                                        
     #1%                                                                        
     \b@nary{\no@}}                                                             
                                                                                
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
%%% Fibonacci trees                                                        %%%  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
                                                                                
% \f@b expands to the description of a Fibonacci tree                           
% of height \f@bht.                                                             
                                                                                
\newcount\f@bht                                                                 
                                                                                
\def\f@b{% draws a Fibonacci tree of depth #1                                   
     \ifnum\f@bht>1                                                             
           \advance\f@bht by-\@ne\f@b\advance\f@bht by\@ne                      
           \advance\f@bht by-\tw@\f@b\advance\f@bht by\tw@                      
           \ifunn@des\node{\unary}                                              
                  \fi                                                           
           \node{\lefttop}                                                      
      \else\ifnum\f@bht=1                                                       
                 \node{\external\le@f}                                          
                 \node{\external\le@f}                                          
                 \node{}                                                        
            \else\node{\external\le@f}                                          
              \fi                                                               
        \fi}                                                                    
                                                                                
\newif\ifunn@des                                                                
                                                                                
\let\unarynodes\unn@destrue                                                     
\def\hght#1{\f@bht=#1\relax}                                                    
                                                                                
\def\fibonacci#1{%                                                              
     \hght{0}\unn@desfalse\circleleaves                                         
     #1%                                                                        
     \f@b}                                                                      
                                                                                
                                                                                
%                                                                               
