The asymptotic number of different rooted trees of a tree

Abstract

Let $\mathcal{T}_n$ be the set of trees with $n$ vertices. Suppose that each tree in $\mathcal{T}_n$ is equally likely. We show that the number of different rooted trees of a tree equals $(\mu_r+o(1))n$ for almost every tree of $\mathcal{T}_n$, where $\mu_r$ is a constant. As an application, we show that the number of any given pattern in $\mathcal{T}_n$ is also asymptotically normally distributed with mean $\sim \mu_M n$ and variance $\sim \sigma_M n$, where $\mu_M, \sigma_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.