We consider the model of phylogenetic trees in which every node of the
tree is an observed, binary random variable and the transition
probabilities are given by the same matrix on each edge of the tree.
The ideal of invariants of this model is a toric ideal in
$C[p_{i_1...i_n}]$. We are able to compute the Gr\"obner basis and
minimal generating set for this ideal for trees with up to 11 nodes.
These are the first nontrivial Gr\"obner bases calculations in
2^11 = 2048 indeterminates. We conjecture that there is a
quadratic Gr\"obner basis for binary trees, but that generators of
degree $n$ are required for some trees with $n$ nodes. The polytopes
associated with these toric ideals display interesting finiteness
properties. We describe the polytope for an infinite family of binary
trees and conjecture (based on extensive computations) that there is a
universal bound on the number of vertices of the polytope of a binary
tree.
