Survival Probabilities for N-ary Subtrees on a Galton-Watson Family Tree

TL;DR

This paper investigates the rate at which N-ary subtrees of finite height vanish in Galton-Watson family trees that do not contain infinite N-ary subtrees, providing insights into the structure and extinction probabilities of such trees.

Contribution

It introduces a new analysis of the decay rate of finite N-ary subtrees in Galton-Watson trees lacking infinite N-ary structures.

Findings

Quantifies the decay rate of N-ary subtrees as height increases

Provides conditions for the absence of infinite N-ary subtrees

Enhances understanding of the structural properties of Galton-Watson trees

Abstract

The family tree of a Galton-Watson branching process may contain N-ary subtrees, i.e. subtrees whose vertices have at least N>0 children. For family trees without infinite N-ary subtrees, we study how fast N-ary subtrees of height t disappear as t goes to infinity.