THE CODING OF VARIOUS KINDS OF UNLABELED TREES

Abstract

This chapter focuses on the coding of various kinds of unlabeled trees. A coding procedure is a mapping c: G →X such that the two graphs in G map onto the same element of X, only if they are isomorphic. In practice, the set X is usually the set of all strings of symbols of some kind. Any process for obtaining from a valid code x a graph whose code is x, that is, a representative of the corresponding isomorphism class will be called a decoding procedure. Although the trees have all been unlabeled, it is clear that if a means of drawing a tree in the plane in some standard way then it is no problem to choose a method for labeling the nodes of the tree in a manner which depends only on the isomorphism class of the tree. There are many ways in which this choice can be made, and a canonical labeling can be obtained. A simple induction argument shows that the canonical labeling thus obtained is precisely that given by the decoding algorithm, Algorithm 1. This suggests that the code for a tree could be derived from a walk around the tree, and this is so.