OSF-Builder: A new tool for constructing and representing evolutionary histories involving introgression.

Introgression is an evolutionary process which provides an important source of innovation for evolution. Although various methods have been used to detect introgression, very few methods are currently available for constructing evolutionary histories involving introgression. In this article, we propose a new method for constructing such evolutionary histories whose starting point is a species forest (consisting of a collection of lineage trees, usually arising as a collection of clades or monophyletic groups in a species tree), and a gene tree for a specific allele of interest, or allele tree for short. Our method is based on representing introgression in terms of a certain "overlay" of the allele tree over the lineage trees, called an overlaid species forest (OSF). OSFs are similar to phylogenetic networks although a key difference is that they typically have multiple roots because each monophyletic group in the species tree has a different point of origin. Employing a new model for introgression, we derive an efficient algorithm for building OSFs called OSF-Builder that is guaranteed to return an optimal OSF in the sense that the number of potential introgression events is minimized. As well as using simulations to assess the performance of OSF-Builder, we illustrate its use on a butterfly data set in which introgression has been previously inferred. The OSF-Builder software is available for download from https://www.uea.ac.uk/computing/software/OSF-Builder.