On a general weight of trees

Abstract

We define a general weight of the nodes of a given tree T; it depends on the structure of the subtrees of a node, on the number of interior and exterior nodes of these subtrees and on three weight functions defined on the degrees of the nodes appearing in T. Choosing particular weight functions, the weight of the root of the tree is equal to its internal path length, to its external path length, to its internal degree path length, to its external degree path length, to its number of nodes of some degree'r, etc. For a simply generated family of rooted planar trees (e.g. all trees defined by restrictions on the set of allowed node degrees), we shall derive a general approach to the computation of the average weight of a tree T ε with n nodes and m leaves for arbitrary weight functions, on the assumption that all these trees are equally likely. This general result implies exact and asymptotic formulas for the average weight of a tree T ε with n nodes for arbitrary weight functions satisfying particular conditions. Furthermore, this approach enables us to derive explicit and asymptotic expressions for the different types of average path lengths of a tree T ε with n nodes and of all ordered trees with n nodes and m leaves.