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In the tropical limit of matrix KP-II solitons, their support at fixed timeis a planar graph with "polarizations" attached to its linear parts. In thiswork we explore a subclass of soliton solutions whose tropical limit graph hasthe form of a rooted and generically binary tree, as well as solutions with alimit graph consisting of two relatively inverted such rooted tree graphs. Thedistribution of polarizations over the constituting lines of the graph is fullydetermined by a parameter-dependent binary operation and a (in generalnon-linear) Yang-Baxter map, which in the vector KP case becomes linear, henceis given by an R-matrix. The parameter-dependence of the binary operation leadsto a solution of the pentagon equation, which exhibits a certain relation withthe Rogers dilogarithm via a solution of the hexagon equation, the next memberin the family of polygon equations. A generalization of the R-matrix, obtainedin the vector KP case, is found to also solve a pentagon equation. Acorresponding local version of the latter then leads to a new solution of thehexagon equation.

Matrix KP: tropical limit, Yang-Baxter and pentagon maps