The asymptotic number of different rooted trees of a tree

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 $(μ_r+o(1))n$ for almost every tree of $\mathcal{T}_n$, where $μ_r$ 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 $\sim μ_M n$ and variance $\sim σ_M n$, where $μ_M, σ_M$ are some constants related to the given pattern. This solves an open question claimed in Kok's thesis.

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