Topology and inference for multi-type Yule trees

Abstract

We introduce two models for multi-type random trees motivated by studies of trait dependence in the evolution of species. Our discrete time model, the multi-type ERM tree, is a generalization of Markov propagation models on a random tree generated by a binary search or `equal rates Markov' mechanism. Our continuous time model, the multi-type Yule tree with mutations, is a multi-type generalization of the tree generated by a pure birth or Yule process. We study type dependent topological properties of these two random tree models. We derive asymptotic results that allow one to infer model parameters from data on types at the leaves and at branch-points that are one step away from the leaves.