Scaling limits of Markov branching trees and Galton-Watson trees conditioned on the number of vertices with out-degree in a given set

TL;DR

This paper extends the understanding of scaling limits for Markov branching and Galton-Watson trees conditioned on specific out-degree node counts, showing convergence to the Brownian continuum random tree.

Contribution

It generalizes previous results to include trees conditioned on out-degree sets and introduces a generalized Otter-Dwass formula for this purpose.

Findings

Scaling limits of Markov branching trees are characterized.

Finite variance Galton-Watson trees conditioned on out-degree sets converge to the Brownian CRT.

A generalized Otter-Dwass formula is developed for these analyses.

Abstract

We generalize recent results of Haas and Miermont to obtain scaling limits of Markov branching trees whose size is specified by the number of nodes whose out-degree lies in a given set. We then show that this implies that the scaling limit of finite variance Galton-Watson trees condition on the number of nodes whose out-degree lies in a given set is the Brownian continuum random tree. The key to this is a generalization of the classical Otter-Dwass formula.