Branch merging on continuum trees with applications to regenerative tree growth

TL;DR

This paper introduces a novel family of branch merging operations on continuum trees, demonstrating their invariance properties and applications to regenerative tree growth, with detailed exploration of Ford CRTs and related processes.

Contribution

It presents new branch merging operations on continuum trees, including Ford CRTs, based on spinal decompositions and regenerative processes, expanding understanding of tree invariance and growth.

Findings

Ford CRTs are distributionally invariant under the new operations

The branch merging operation is new even for the Brownian CRT

Provides an alternative approach to leaf embedding in Ford CRTs

Abstract

We introduce a family of branch merging operations on continuum trees and show that Ford CRTs are distributionally invariant. This operation is new even in the special case of the Brownian CRT, which we explore in more detail. The operations are based on spinal decompositions and a regenerativity preserving merging procedure of $(\alpha, \theta)$-strings of beads, that is, random intervals $[0, L_{\alpha, \theta}]$ equipped with a random discrete measure $dL^{-1}$ arising in the limit of ordered $(\alpha, \theta)$-Chinese restaurant processes as introduced recently by Pitman and Winkel. Indeed, we iterate the branch merging operation recursively and give an alternative approach to the leaf embedding problem on Ford CRTs related to $(\alpha, 2-\alpha)$-regenerative tree growth processes.