The Distribution of Patterns in Random Trees

Let Tn denote the set of unrooted labeled trees of size n and let M be a particular (finite, unlabeled) tree. Assuming that every tree of Tn is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of M as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to μn and σn, respectively, where the constants μ > 0 and σ ≥ 0 are computable.