Recall that when replacing trees by dual polygons, one can distinguish the type
of the associahedron face by the kind of diagonals for the polygon. For example, the K4 Stasheff polytope has 6 pentagonal faces and 3 squares. These are distinguished by the chorded hexagons
where a diagonal that splits a hexagon in two corresponds to a square. This shows how pentagons may be paired, by taking dual diagonals, but squares are at best self dual. Labelled trees may be replaced by labelled polygons.
The description of trees as clusters of polygons, used by Devadoss
in tiling moduli spaces, is better known to category theorists as the theory of 2-opetopes
. The dimension 2 describes the planar nature of polygons, but this may be generalised. On that note, David Corfield
points out a wonderful new paper on the arxiv.