A new family of Markov branching trees: the alpha-gamma model

TL;DR

This paper introduces the alpha-gamma model, a new two-parameter family of discrete fragmentation trees that generalizes existing models and connects to stable continuum random trees, providing detailed splitting rules and limiting behavior analysis.

Contribution

It presents the alpha-gamma trees, extending Ford's alpha model to multifurcating trees and linking discrete and continuum random trees with comprehensive mathematical properties.

Findings

Derived splitting rules for alpha-gamma trees

Established dislocation measures in ranked and biased order

Analyzed the limiting behavior of the new tree family

Abstract

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour.