Hilbert Polynomial of the Kimura 3-Parameter Model

In [2] Buczynska and Wisniewski showed that the Hilbert polynomial of the algebraic variety associated to the Jukes-Cantor binary model on a trivalent tree depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other group-based models. The Jukes-Cantor binary model has Z(2) as the underlying group. We consider the Kimura 3-parameter model with Z(2) x Z(2) as the underlying group. We show that the generalization of the statement about the Hilbert polynomials to the Kimura 3-parameter model is not possible as the Hilbert polynomial depends on the shape of a trivalent tree.