Abstract: As the result of Mendelian inheritance, individuals within a population share a genealogical history (i.e. a “gene tree”) with each other at every locus in the genome. The coalescent is a mathematical model that helps us to understand what these genealogies look like within a species, and the multispecies coalescent extends this model between species by considering coalescence in ancestral populations. In this talk, I will discuss two further extensions: one that considers genealogical relationships among different loci related by duplication events (with possible losses as well), and one that considers genealogical relationships between unrelated species driven by introgression. I present an approach for calculating the number of possible gene trees in the presence of duplication, loss, and coalescence, a number that increases factorially with the number of species considered. Following this, I show how the same general approach can be used to calculate numbers of gene trees and probabilities of gene trees when introgression is also acting. Such a model is useful for both detecting the presence of introgression and determining the number of unique introgression events in a species tree.