The cut-tree of large trees with small heights

TL;DR

This paper studies the limiting behavior of the cut-tree process for large finite trees with small height, establishing convergence criteria and extending previous results to various tree models.

Contribution

It provides a general criterion for the convergence of rescaled cut-trees of small-height trees and extends existing results to new classes of random trees.

Findings

Convergence of rescaled cut-trees to an interval in Gromov-Prohorov topology.

Extension of Bertoin's results to new tree models.

Generalization of vertex isolation results in random trees.

Abstract

We destroy a finite tree of size $n$ by cutting its edges one after the other and in uniform random order. Informally, the associated cut-tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut-tree in the Gromov-Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has a small height of order $o(\sqrt{n})$. In particular, we consider uniform random recursive trees, binary search trees, scale-free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin for the cut-tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut-tree of other classes of trees.