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 non-trivial 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.