Mass-structure of weighted real trees

TL;DR

This paper introduces a framework for understanding the interaction between structure and mass in weighted real trees, defining equivalence classes, and proposing a new family of trees called interval partition trees as canonical representatives.

Contribution

It develops a notion of mass-structure equivalence for weighted real trees and introduces interval partition trees as canonical representatives of these classes.

Findings

Mass-structure equivalence characterized by discrete hierarchies

Interval partition trees serve as natural representatives

Framework links tree structure with probabilistic hierarchies

Abstract

Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(\mathcal{T},d,r,p)$, where $(\mathcal{T},d)$ is a tree-like metric space, $r\in\mathcal{T}$ is a distinguished root, and $p$ is a probability measure on this space. The underlying branching structure is carried implicitly in the metric $d$. We explore various ways of describing the interaction between branching structure and mass in $(\mathcal{T},d,r,p)$ in a way that depends on $d$ only by way of this branching structure. We introduce a notion of mass-structure equivalence and show that two rooted, weighted $\mathbb{R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingman's paintbox, have the same distribution. We introduce a family of trees, called "interval partition trees" that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.