Branch Length Transforms using Optimal Tree Metric Matching.

The abundant discordance between evolutionary relationships across the genome has rekindled interest in ways of comparing and averaging trees on a shared leaf set. However, most attempts at reconciling trees have focused on tree topology, producing metrics for comparing topologies and methods for computing median tree topologies. Using branch lengths, however, has been more elusive, due to several challenges. Species tree branch lengths can be measured in many units, often different from gene trees. Moreover, rates of evolution change across the genome, the species tree, and specific branches of gene trees. These factors compound the stochasticity of coalescence times. Thus, branch lengths are highly heterogeneous across both the genome and the tree. For many downstream applications in phylogenomic analyses, branch lengths are as important as the topology, and yet, existing tools to compare and combine weighted trees are limited. In this paper, we make progress on the question of mapping one tree to another, incorporating both topology and branch length. We define a series of computational problems to formalize finding the best transformation of one tree to another while maintaining its topology and other constraints. We show that all these problems can be solved in quadratic time and memory using a linear algebraic formulation coupled with dynamic programming preprocessing. Our formulations lead to convex optimization problems, with efficient and theoretically optimal solutions. While many applications can be imagined for this framework, we apply it to measure species tree branch lengths in the unit of the expected number of substitutions per site while allowing divergence from ultrametricity across the tree. In these applications, our method matches or surpasses other methods designed directly for solving those problems. Thus, our approach provides a versatile toolkit that finds applications in similar evolutionary questions. Code availability:The software is available at https://github.com/shayesteh99/TCMM.git . Data availability:Data are available on Github https://github.com/shayesteh99/TCMM-Data.git .