  In the tropical limit of matrix KP-II solitons, their support at fixed time
is a planar graph with "polarizations" attached to its linear parts. In this
work we explore a subclass of soliton solutions whose tropical limit graph has
the form of a rooted and generically binary tree, as well as solutions with a
limit graph consisting of two relatively inverted such rooted tree graphs. The
distribution of polarizations over the constituting lines of the graph is fully
determined by a parameter-dependent binary operation and a (in general
non-linear) Yang-Baxter map, which in the vector KP case becomes linear, hence
is given by an R-matrix. The parameter-dependence of the binary operation leads
to a solution of the pentagon equation, which exhibits a certain relation with
the Rogers dilogarithm via a solution of the hexagon equation, the next member
in the family of polygon equations. A generalization of the R-matrix, obtained
in the vector KP case, is found to also solve a pentagon equation. A
corresponding local version of the latter then leads to a new solution of the
hexagon equation.
