  Speyer and Sturmfels [SpSt] associated Gr\"obner toric degenerations
$\mathrm{Gr}_2(\C^n)^{\tree}$ of $\mathrm{Gr}_2(\C^n)$ to each trivalent tree
$\tree$ with $n$ leaves. These degenerations induce toric degenerations
$M_{\br}^{\tree}$ of $M_{\br}$, the space of $n$ ordered, weighted (by $\br$)
points on the projective line. Our goal in this paper is to give a geometric
(Euclidean polygon) description of the toric fibers as stratified symplectic
spaces and describe the action of the compact part of the torus as "bendings of
polygons." We prove the conjecture of Foth and Hu [FH] that the toric fibers
are homeomorphic to the spaces defined by Kamiyama and Yoshida [KY].
