How does a map from permutahedra to associahedra factor through multiplihedra?

Let $$P_i$$ denote permutahedra, $$K_i$$ associahedra and $$J_i$$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $$p_i: P_i \to K_{i+1}$$ which factors as $$P_i \to J_i \to K_{i+1}$$. The projection is as follows: we index faces of $$P_i$$ with leveled planar binary trees with $$i+1$$ leaves, and we index faces of $$K_{i+1}$$ with planar binary trees with $$i+1$$ leaves; so $$p_i$$ is by disregarding levels.

However, one labelling of $$J_i$$'s faces that I know is by appropriately painted planar binary trees with $$i$$ leaves, not $$i+1$$ (as in Forcey https://arxiv.org/abs/0706.3226). So I am having a problem understanding the factorization above. How does one see painted trees with $$i$$ leaves as equivalence classes of leveled trees with $$i+1$$ leaves?

(In case $$i=3$$, $$P_3$$ and $$J_3$$ are both hexagons, so the equivalence classes are just with one element - there should be a bijection, but which bijection?)

• Maybe use the level of the unique inner vertex under the rightmost leaf of the leveled tree to fix the painting height ? Oct 21, 2020 at 19:56