Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, respectively the study of mutations of organisms. Both theoretically and in terms of applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree 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 phylogenetic degree relies on computing the volume of their associated polytopes $P_{G,n}$. We apply combinatorial methods and we give concrete formulas for them.