Tropical $k$-means clustering for phylogenetic trees

The asymmetric tropical distance is a distance measure on the tropical torus ${\mathbb{R}}^n\mathrm{/}\mathbb{R}\mathbf{1}$ and in particular on the Bergman fan $B(K_N)\subseteq {\mathbb{R}}^{\left(\genfrac{}{}{0px}{}{N}{2}\right)}\mathrm{/}\mathbb{R}\mathbf{1}$ of the complete graphical matroid. In this paper, we define and analyse a clustering algorithm for equidistant phylogenetic trees based on this distance, using the correspondence between $B(K_N)$ and the space of equidistant trees with $N$ leaves.