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 [16]. Two sibling subtrees are placed as close together as possible, during a postorder traversal, as follows. At each node 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 [15]. 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.