Invariant Galton-Watson trees: metric properties and attraction with respect to generalized dynamical pruning

TL;DR

This paper studies invariant Galton-Watson trees, showing they are unique attractors under certain pruning operations and deriving their key metric distributions.

Contribution

It establishes the uniqueness of invariant Galton-Watson measures as attractors under generalized dynamical pruning and characterizes their metric properties.

Findings

Invariant Galton-Watson measures are the only attractors under generalized dynamical pruning.

Distributions of height, length, and size of IGW trees are explicitly derived.

IGW trees exhibit specific metric properties under critical conditions.

Abstract

Invariant Galton-Watson (IGW) tree measures is a one-parameter family of critical Galton-Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are the only attractors of critical Galton-Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.