Probabilistic Models for the (sub)Tree(s) of Life

TL;DR

This paper reviews mathematical models of random trees in evolutionary biology, focusing on stochastic tree shapes, real trees, and coalescent trees, with applications to species diversification, conservation, and epidemiology.

Contribution

It introduces a unified framework for understanding various random tree models, including the $eta$-family and comb metric representations, with recent biological applications.

Findings

Representation of coalescent trees via comb metric

Application of coalescent point processes to biological inference

Analysis of random tree models in evolutionary biology

Abstract

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $\beta$-family of Aldous' branching models. We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered. We then focus on the reduced tree, or coalescent tree, which is the tree spanned by individuals/species alive at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes. We end up displaying some recent biological applications of coalescent point processes to the inference of species diversification, to conservation biology and to epidemiology.