On general subtrees of a conditioned Galton-Watson tree

TL;DR

This paper proves that the frequency of a specific rooted subtree in a conditioned Galton-Watson tree converges to a deterministic value under minimal offspring distribution moments.

Contribution

It establishes a law of large numbers for subtree counts in conditioned Galton-Watson trees with minimal moment assumptions.

Findings

Number of subtree copies satisfies a law of large numbers.

Convergence holds under minimal moment conditions.

Provides a probabilistic understanding of subtree distribution.

Abstract

We show that the number of copies of a given rooted tree in a conditioned Galton-Watson tree satisfies a law of large numbers under a minimal moment condition on the offspring distribution.