# The isometry group of phylogenetic tree space is $S_n$

## TL;DR

This paper proves that the isometry group of phylogenetic tree space, a metric space of all trees with a fixed leaf set, is the symmetric group on n elements, impacting phylogenetic analysis methods.

## Contribution

It establishes that the isometry group of phylogenetic tree space is the symmetric group, a novel result linking geometry and combinatorics in phylogenetics.

## Key findings

- Isometry group is the symmetric group $S_n$
- Implications for distance-based phylogenetic analysis
- Clarifies symmetry properties of tree space

## Abstract

A phylogenetic tree is an acyclic graph with distinctly labeled leaves, whose internal edges have a positive weight. Given a set of n leaves, the collection of all phylogenetic trees with this leaf set can be assembled into a metric cube complex known as phylogenetic tree space, or Billera-Holmes-Vogtmann tree space. In this largely combinatorial paper, we show that the isometry group of this space is the symmetric group on n elements. This fact is relevant to distance-based analyses of phylogenetic tree sets.