Tropical Geometric Variation of Phylogenetic Tree Shapes

TL;DR

This paper explores the tropical geometric interpretation of phylogenetic tree shapes, providing new theoretical insights and algorithms that improve computational efficiency over existing methods like BHV space.

Contribution

It introduces a combinatorial framework for analyzing tree topologies along tropical line segments and presents a more efficient algorithm for computing these segments.

Findings

Tropical line segments can describe tree topology variations without crossing the origin.

The proposed algorithm is computationally faster than existing BHV geodesic methods.

Tropical line segments exhibit more nuanced behavior compared to BHV geodesics.

Abstract

We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree's branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.