On general subtrees of a conditioned Galton–Watson tree

Svante Janson

doi:10.1214/21-ecp392

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.

Highlights

Let Tn be a random conditioned Galton–Watson tree with n nodes, defined by an offspring distribution ξ with mean E ξ = 1, and let t be a fixed ordered rooted tree
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
We are interested in the number of copies of t as a subtree of Tn, which we denote by Nt(Tn)

Summary

Introduction

Let Tn be a random conditioned Galton–Watson tree with n nodes, defined by an offspring distribution ξ with mean E ξ = 1, and let t be a fixed ordered rooted tree. Only uniformly random labelled trees are considered in [2], but the proofs extend to suitable more general conditioned Galton– Watson trees, as remarked in [2] and shown explicitly in [12; 13].) It was shown in Chyzak, Drmota, Klausner and Kok [2] that the number of occurences of such a pattern is asymptotically normal, with asymptotic mean and variance both of the order n (except that the variance might be smaller in at least one exceptional degenerate case), which of cource entails a law of large numbers. Quite different from the analysis of generating functions in [2]

Notation

Examples

Asymptotic normality?