The stable trees are nested

TL;DR

This paper demonstrates that all stable trees with parameters between 1 and 2 can be embedded within each other as nested structures, using pruning and fragmentation techniques, revealing a hierarchical organization of stable trees.

Contribution

It introduces a method to construct all stable trees as nested within each other, using pruning procedures and recursive convergence, providing new insights into their hierarchical structure.

Findings

All a-stable trees can be embedded within a'-stable trees for 1 < a < a' ≤ 2.

Nested stable trees can be explicitly constructed via pruning or fragmentation modifications.

The recursive construction converges almost surely to stable trees, ensuring the validity of the nesting approach.

Abstract

We show that we can construct simultaneously all the stable trees as a nested family. More precisely, if $1 < a < a' \leq 2$ we prove that hidden inside any a-stable we can find a version of an a'-stable tree rescaled by an independent Mittag-Leffler type distribution. This tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it. Our proofs are based on a recursive construction due to Marchal which is proved to converge almost surely towards a stable tree.