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.

72 pages, 15 figures. Lecture notes of a mini-course given at the XIX Brazilian School of Probability (aug 2015), random tree, tree shape, real tree, reduced tree, branching process, coalescent, comb, phylogenetics, population dynamics, population genetics. To appear in the Brazilian Journal of Probability and Statistics

References in corpus (6)

• Asymptotic genealogy of a critical branching process
• Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models
• The allelic partition for coalescent point processes
• The coding of compact real trees by real valued functions
• Scaling limits of Markov-Branching trees and applications
• Predicting the loss of phylogenetic diversity under non-stationary diversification models

Cited by in corpus (6)

• Combinatorial and stochastic properties of ranked tree-child networks
• Trees within trees: Simple nested coalescents
• Totally Ordered Measured Trees and Splitting Trees with Infinite Variation
• The split-and-drift random graph, a null model for speciation
• Revisiting Shao and Sokal's $B_2$ index of phylogenetic balance
• Markovian tricks for non-Markovian trees: contour process, extinction and scaling limits