Random ultrametric trees and applications*

Abstract

Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree.

Keywords

• random tree
• real tree
• reduced tree
• coalescent point process
• branching process
• random point measure
• allelic partition
• regenerative set
• coalescent
• comb
• phylogenetics
• population dynamics
• population genetics