M Theory Lesson 293
A basic three leaf tree is what labels the associator edge. In other words, an associator edge is like two four point diagrams connected by a single three point diagram. This is backwards (or rather, Poincare dual) to the situation of interest in twistor QFT, which is where a four point vertex may factorize into pairs of three point vertices.
As it happens, there is such a thing as a dual square, where points become edges and edges points. In this context, roughly speaking, the square would be called a Hodges diagram. The vertices on a Hodges diagram stand for twistor variables in a massless field theory. Poincare duality is a simple enough idea, so massless particle physics really does see associahedra.
That's category theory, folks. You can keep saying that category theory has nothing to do with physics. But Nature does not agree with you.