Algorithms for Unrooted Gene Trees with Polytomies

Gene tree reconciliation is a method to reconcile gene trees that are confounded by complex histories of gene duplications with a provided species tree. The trees involved are required to be rooted and full binary. Reconciling gene trees allows not only to identify and study such histories for gene families, but is also the base for several higher level applications including the estimation of species trees from gene trees when when duplications are involved. However, gene tree reconciliation can not handle common gene trees that are unrooted and non-binary trees. This limitation severely limits the applicability of gene tree reconciliation, since common trees are frequently inferred by phylogenetic methods in practice. Here, we describe a linear time reconciliation of a given common gene tree with a given species tree that can be non-binary. Then, we extend this reconciliation by seeking an optimal reconciliation under all rooted and binary refinements of the common gene tree with the given species tree. Finally, we describe a polynomial time algorithm that computes such an optimal refinement for the case when the species tree is full binary.