Optimal Completion of Incomplete Gene Trees in Polynomial Time Using OCTAL

Abstract

Here we introduce the Optimal Tree Completion Problem, a general optimization problem that involves completing an unrooted binary tree (i.e., adding missing leaves) so as to minimize its distance from a reference tree on a superset of the leaves. More formally, given a pair of unrooted binary trees (T,t) where T has leaf set S and t has leaf set R, a subset of S, we wish to add all the leaves from S R to t so as to produce a new tree t' on leaf set S that has the minimum distance to T. We show that when the distance is defined by the Robinson-Foulds (RF) distance, an optimal solution can be found in polynomial time. We also present OCTAL, an algorithm that solves this RF Optimal Tree Completion Problem exactly in quadratic time. We report on a simulation study where we complete estimated gene trees using a reference tree that is based on a species tree estimated from a multi-locus dataset. OCTAL produces completed gene trees that are closer to the true gene trees than an existing heuristic approach, but the accuracy of the completed gene trees computed by OCTAL depends on how topologically similar the estimated species tree is to the true gene tree. Hence, under conditions with relatively low gene tree heterogeneity, OCTAL can be used to provide highly accurate completions of estimated gene trees. We close with a discussion of future research.