  We address the problem of studying the toric ideals of phylogenetic
invariants for a general group-based model on an arbitrary claw tree. We focus
on the group $\mathbb Z_2$ and choose a natural recursive approach that extends
to other groups. The study of the lattice associated with each phylogenetic
ideal produces a list of circuits that generate the corresponding lattice
ideal. In addition, we describe explicitly a quadratic lexicographic
Gr\"{o}bner basis of the toric ideal of invariants for the claw tree on an
arbitrary number of leaves. Combined with a result of Sturmfels and Sullivant,
this implies that the phylogenetic ideal of every tree for the group $\mathbb
Z_2$ has a quadratic Gr\"{o}bner basis. Hence, the coordinate ring of the toric
variety is a Koszul algebra.
