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

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.