Tree Topologies along a Tropical Line Segment

TL;DR

This paper investigates the combinatorial properties of tree topologies along tropical line segments in tropical geometry, revealing probabilistic and topological characteristics relevant to phylogenetic trees.

Contribution

It provides new insights into the structure of tropical line segments between phylogenetic trees, especially regarding topology changes and probabilities of passing through the star tree.

Findings

Probability of passing through the star tree is zero for more than four leaves.

Tree topology remains unchanged if trees differ by one NNI move.

Properties of tropical line segments help interpret statistical models in tree spaces.

Abstract

Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.