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 $G\in \{{\mathbb{Z}}_2,{\mathbb{Z}}_3,{\mathbb{Z}}_2\times {\mathbb{Z}}_2,{\mathbb{Z}}_4,{\mathbb{Z}}_5,{\mathbb{Z}}_7\}$.