The Distribution of Patterns in Random Trees

Abstract

Let $\Tn{}$ denote the set of unrooted labelled trees of size n and let ℳ be a particular (finite, unlabelled) 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 is asymptotically normal with mean value and variance asymptotically equivalent to μn and σ2n, respectively, where the constants μ>0 and σ≥0 are computable.