Structural Complexity of Random Binary Trees

For each positive integer n, let T-n be a random rooted full binary tree having 2n-1 vertices. We can view H(T-n), the entropy of T-n, as a measure of the structural complexity of tree T-n in the sense that approximately H(T-n) bits suffice to construct T-n. We analyze some random binary tree sequences (T-n : n = 1, 2,...) for which the normalized entropies H(T-n)/n converge to a limit as n -> infinity, as well as some other sequences (T-n) in which the normalized entropies fail to converge.