Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions

TL;DR

This paper introduces a new method for constructing binary self-similar continuum trees using an extended Chinese Restaurant Process, linking discrete Ford trees to their continuum limits through regenerative partitions.

Contribution

It develops a novel approach to establish limits of non-sampling consistent Markov branching trees via regenerative interval partitions and urn models.

Findings

Explicit embedding of Ford's alpha model trees in continuum trees

New approach to prove existence of compact limiting trees

Connection between discrete and continuum tree models

Abstract

We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions.