Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in studying the algebraic properties of the phylogenetic varieties coming from these models. In this talk, we discuss the algebraic degrees of these varieties. We will present concrete results on the algebraic degrees of the variety $X_{G, n}$ with $G\in\{\mathbb{Z}_2,\mathbb{Z}_2\times \mathbb{Z}_2, \mathbb{Z}_3\}$ and any $n$-claw tree. As these varieties are toric, computing their algebraic degree relies on computing the volume of their associated polytopes $P_{G,n}$. We will present algebraic and combinatorial methods used in our work. This talk is based on joint works with Martin Vodi\v{c}ka.