Random tree recursions: which fixed points correspond to tangible sets of trees?

TL;DR

This paper investigates fixed points of recursive properties in Galton-Watson trees, classifying solutions based on their probabilistic interpretations using spine decompositions and Boolean function analysis.

Contribution

It introduces a framework for understanding fixed points of recursive tree properties and classifies solutions by their probabilistic significance.

Findings

Multiple fixed points can exist for the recursive property.

Not all fixed points correspond to actual probabilities of tree properties.

The framework distinguishes between meaningful probabilistic solutions and mathematical artifacts.

Abstract

Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.