Classification of normal phylogenetic varieties for tripods

We provide a complete classification of normal phylogenetic varieties coming from tripods and, more generally, from trivalent trees. Let G be an abelian group. We prove that the group-based phylogenetic variety X_{G,\mathcal{T}} , for any trivalent tree \mathcal{T} , is projectively normal if and only if G\in \{\mathbb{Z}_{2}, \mathbb{Z}_{3}, \mathbb{Z}_{2}\times\mathbb{Z}_{2}, \mathbb{Z}_{4}, \mathbb{Z}_{5}, \mathbb{Z}_{7}\} .