The Counting and Coding of Trees of Fixed Diameter

Abstract

Recursive and asymptotic formulas are obtained for enumerating various classes of trees under the constraints of a fixed diameter and an upper bound on the degrees of their points. Such formulas are of considerable interest in chemical applications of graph theory. The methods used are also applied to the related problem of coding trees, and algorithms are given for assigning unique integer codes to each member of a class of trees. The algorithms lead to sequences such that each tree is preceded by its subtrees, i.e., no tree has a smaller integer code than any of its subtrees. This is a useful feature for chemical additivity schemes. Equations are given for coding and decoding which render the code especially amenable to computer applications. The process of decoding, i.e., finding a tree (topology) given its integer code, amounts to solving a diophantine equation, under inequality constraints. The existence of a unique solution is guaranteed by the graph-theoretical construction.