Does freezing impede the growth of random recursive trees?

TL;DR

This paper investigates how freezing vertices affects the height of random recursive trees, revealing that freezing does not significantly reduce height and removing attachment can increase it.

Contribution

It introduces a model of uniform attachment with freezing and analyzes the impact of these modifications on tree height using coupling methods.

Findings

Removing an attachment step can increase expected tree height.

Freezing vertices does not substantially decrease the height.

Coupling arguments are used to analyze the model.

Abstract

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.