# Random ultrametric trees and applications

## TL;DR

This paper explores the structure of ultrametric trees, introduces a comb metric representation for their boundaries, and discusses applications in modeling population genetics and genealogies.

## Contribution

It presents a novel representation of ultrametric tree boundaries using comb metrics and reviews applications in population genetics modeling.

## Key findings

- Boundary of ultrametric trees can be represented via comb metrics
- Random combs can model genetic structures in populations
- Applications include analyzing neutral mutations in genealogies

## 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.