  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.
