Structural complexity of random binary trees

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