  We consider the phylogenetic tree model in which every node of the tree is
observed and binary and the transitions are given by the same matrix on each
edge of the tree. We are able to compute the Grobner basis and Markov basis of
the toric ideal of invariants for trees with up to 11 nodes. These are perhaps
the first non-trivial Grobner bases calculations in 2^11 indeterminates. We
conjecture that there is a quadratic Grobner basis for binary trees. Finally,
we give a explicit description of the polytope associated to this toric ideal
for an infinite family of binary trees and conjecture that there is a universal
bound on the number of vertices of this polytope for binary trees.
