Recursive functions on conditional Galton--Watson trees

TL;DR

This paper investigates the limiting behavior of recursive functions on random Galton--Watson trees with finite state values, focusing on the root's value when leaf values are randomly assigned.

Contribution

It introduces a framework for analyzing recursive functions on Galton--Watson trees with finite state spaces, exploring their limit behavior as tree size grows.

Findings

Describes the limit distribution of root values in large random trees.

Establishes conditions for convergence of recursive functions.

Provides a foundation for analyzing recursive processes on random trees.

Abstract

A recursive function on a tree is a function in which each leaf has a given value, and each internal node has a value equal to a function of the number of children, the values of the children, and possibly an explicitly specified random element $U$. The value of the root is the key quantity of interest in general. In this first study, all node values and function values are in a finite set $S$. In this note, we describe the limit behavior when the leaf values are drawn independently from a fixed distribution on $S$, and the tree $T_n$ is a random Galton--Watson tree of size $n$.