Cutting down trees with a Markov chainsaw

TL;DR

This paper offers simplified proofs for the distribution of cuts needed to down a Galton-Watson tree, connecting probabilistic structures with new transformations between Brownian processes.

Contribution

It introduces a coupling method for precise distributional results and a novel reversible transformation between Brownian excursion and bridge.

Findings

Provides nonasymptotic distributional results for random trees

Establishes a new reversible transformation between Brownian processes

Simplifies existing proofs for tree cutting distributions

Abstract

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton-Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny $n$. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton-Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge.