Asymptotic normality for general subtree counts in conditioned Galton--Watson trees

TL;DR

This paper proves that the count of a fixed subtree in conditioned Galton--Watson trees follows an asymptotic normal distribution under certain conditions, confirming a conjecture and highlighting the importance of moment assumptions.

Contribution

It establishes asymptotic normality for subtree counts in conditioned Galton--Watson trees under mild moment conditions, confirming a conjecture and providing counterexamples.

Findings

Subtree counts are asymptotically normal with linear mean and variance.

The distribution is nondegenerate except in special cases.

Violating moment conditions can lead to failure of asymptotic normality.

Abstract

Let $\mathcal{T}$ denote a Galton--Watson tree with offspring distribution $\xi$ satisfying $\mathbb{E}(\xi) = 1$, and let $\mathcal{T}_n$ be the Galton--Watson tree conditioned to have exactly $n$ nodes. We show that, under a mild moment condition on $\xi$, the number of occurrences of a fixed rooted plane tree $\mathbf{t}$ as a general subtree in $\mathcal{T}_n$ is asymptotically normal as $n \to \infty$, with both mean and variance linear in $n$. In addition, we prove that this limiting distribution is nondegenerate except for some special cases where the variance remains bounded. These results confirm a conjecture of Janson in recent work on the same topic. Finally, we present examples showing that if the proposed moment condition on $\xi$ is violated, the conclusion may fail.