# Tropical Optimal Transport and Wasserstein Distances in Phylogenetic Tree Space

### Abstract

We study the problem of optimal transport on phylogenetic tree space from the perspective of tropical geometry, and thus define the Wasserstein-p distances for probability measures in this continuous metric measure space setting. With respect to the tropical metric—a combinatorial metric on the space of phylogenetic trees—the cases of $p=1,2$ are treated in detail, which give an efficient way to compute geodesics and also provide theoretical foundations for geometric insight a statistical framework on tree space. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances, and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport on sets of phylogenetic trees. Several numerical examples are provided.

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