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 rarr infin, as well as some other sequences (T_n) in which the normalized entropies fail to converge.