# Brownian motion tree models are toric

## TL;DR

This paper demonstrates that Felsenstein's Gaussian phylogenetic tree model is a toric variety, providing an algebraic geometric perspective that enables exact maximum likelihood estimation and offers new insights into ultrametric matrix geometry.

## Contribution

It establishes the toric structure of the classical Gaussian phylogenetic model and derives an exact semialgebraic characterization, advancing algebraic methods in phylogenetics.

## Key findings

- Model is a toric variety in concentration matrix space
- Provides exact semialgebraic characterization of the model
- Enables algebraic methods for maximum likelihood estimation

## Abstract

Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.