Non-fringe subtrees in conditioned Galton--Watson trees

TL;DR

This paper investigates the distribution and moments of the number of subtrees in conditioned Galton--Watson trees, revealing a log-normal distribution and exponential scale moments through two distinct analytical methods.

Contribution

It introduces two different methods to analyze the distribution and moments of subtree counts in conditioned Galton--Watson trees, establishing new probabilistic results.

Findings

Logarithm of subtree count follows a Central Limit Law.

Moments of subtree count are of exponential scale.

Provides new insights into the structure of conditioned Galton--Watson trees.

Abstract

We study $S(\mathcal T_{n})$, the number of subtrees in a conditioned Galton--Watson tree of size $n$. With two very different methods, we show that $\log(S(\mathcal T_{n}))$ has a Central Limit Law and that the moments of $S(\mathcal T_{n})$ are of exponential scale.