On the unranked topology of maximally probable ranked gene tree topologies

A ranked tree topology is a tree topology with a temporal ordering of its coalescence events. Under the multispecies coalescent model, we consider ranked gene tree topologies realized along the branches of ranked species trees, where one gene copy is sampled for each species. Previous results have demonstrated that for almost all ranked species tree topologies with at least five species, there exists a set of branch lengths such that the maximally probable ranked gene tree topologies-those generated with the highest probability under the model-do not match the species tree ranked topology. Here, we focus on the agreement of a ranked species tree with its maximally probable ranked gene tree topologies in terms of their unranked topology, that is, disregarding the ordering of the coalescence events. We show that although the set of maximally probable ranked gene tree topologies for a ranked species tree can contain ranked trees with different unranked topologies, at least one of these maximal ranked gene tree topologies must have the same unranked topology as the species tree. Our results contribute to the study of the relationships between gene trees and species trees.