  Equivariant tree models are statistical models used in the reconstruction of
phylogenetic trees from genetic data. Here equivariant refers to a symmetry
group imposed on the root distribution and on the transition matrices in the
model. We prove that if that symmetry group is Abelian, then the Zariski
closures of these models are defined by polynomial equations of bounded degree,
independent of the tree. Moreover, we show that there exists a polynomial-time
membership test for that Zariski closure. This generalises earlier results on
tensors of bounded rank, which correspond to the case where the group is
trivial, and implies a qualitative variant of a quantitative conjecture by
Sturmfels and Sullivant in the case where the group and the alphabet coincide.
Our proofs exploit the symmetries of an infinite-dimensional projective limit
of Abelian star models.
