On path entropy functions for rooted trees Original Research Article

If u is a terminal node of a rooted tree Tn with n terminal nodes, let h(u) = Σf(d(v)) where the sum is over all interior nodes v in the path from the root of Tn to u, d(v) is the out-degree of v, and f is a non-negative cost function. The path entropy function h(Tn) = Σh(u), where the sum is over the n terminal nodes of Tn, is a measure of the complexity of the hierarchical classification scheme represented by Tn. We show, under suitable assumptions, that the expected value of h(Tn) over all trees Tn in certain families of weighted trees is asymptotic to Kn32 where the constant K depends on the family and the cost function f.