On path entropy functions for rooted trees

Abstract If u is a terminal node of a rooted tree T n 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 T n to u , d ( v ) is the out-degree of v , and f is a non-negative cost function. The path entropy function h ( T n ) = Σh ( u ), where the sum is over the n terminal nodes of T n , is a measure of the complexity of the hierarchical classification scheme represented by T n . We show, under suitable assumptions, that the expected value of h ( T n ) over all trees T n in certain families of weighted trees is asymptotic to Kn 3 2 where the constant K depends on the family and the cost function f.