2016/03/31 by Amaury Lambert
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Algorithm #Artificial intelligence #Bayesian Methods and Mixture Models #Binary tree #Biology #Coalescent theory #Combinatorics #Computer science #Diffusion and Search Dynamics #Discrete mathematics #Mathematics #Metric space #Phylogenetic tree #Random binary tree #Random tree #Stochastic processes and statistical mechanics #Topological data analysis #Tree (set theory) #Tree rearrangement #Ultrametric space #math.PR #q-bio.PE
paper · pdf · doi:10.1214/16-bjps320
published as Brazilian Journal of Probability and Statistics 2017, 31(3) 415-475 · 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
arxiv created 2016/07/08 · openalex publication_date 2017/08/01 · arxiv updated 2017/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model species trees. We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the β-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 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.