Presentations of semigroup algebras of weighted trees

We study presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski. These algebras arise as toric degenerations of projective coordinate rings of weight varieties of the Grassmannian of two planes associated to the Plucker embedding, and as toric degenerations of rings of invariants of Cox-Nagata rings.