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 in (J. Eur. Math. Soc. 9:609–635, 2007). These algebras arise as toric degenerations of projective coordinate rings of the moduli of weighted points on the projective line, and projective coordinate rings of the moduli of quasiparabolic semisimple rank two bundles on the projective line.