I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF). For some analytics see https://oeis.org A134685 (LIF for e.g.f.s) related to A134991 [tropical Grassmannian G(2,n)], A133437 (LIF for o.g.f.s) with A033282 (associahedra), and A134264 (LIF using reciprocal) with A001263 (dual to associahedra). (Also you might refer to my earlier questions Q1 and Q2)

Reviewing an old reference in A133437, I found a good way to introduce the question on page 15 of Loday's paper "The Multiple Facets of the Associahedron" in Sec. 6 Inversion of Power Series. He ends with "There exists a short operadic proof of the above formula [LIF essentially] which explicitly involves the parenthesizings [of associahedra], but it would be interesting to find one which involves the topological structure of the associahedron."

I have a vague insight which I stated in my Grobner question. I'm interested in hearing from others about their insights if any. I don't expect a definitive aswer to such a question, nor wish for anyone to become frustrated at the inherent vagueness of the question, so let me restate it simply: Why (vague notion) do the face numbers of the associahedra appear as the coeficients of Lagrange inversion/reversion for an ordinary generating fct. as presented in OEIS A133437?

One viewpoint, for example: I can derive the LIF several ways and relate the methods to rooted trees, but is there an intuitive way to relate the LIF for compositional inversion (which is related to the Legendre transformation/Legendre-Fenchel transform) to the geometry of the associahedra through a geometrical view of optimization via integer programming? Compositional inversion and the Legendre transformation have geometrical interpretations and are related to optimization as discussed by Strang in his book Intro. to Applied Mathematics (see also Mathemagical Forests and references therein in the section A Walk With Lagrange and Legendre). De Loera, Rambau and Leal in Triangulations of Set Points in Sec. 1.2 Optimization and Triangulations discuss connections of secondary polytopes to optimization.

Second viewpoint: Stasheff associahedra are intimately related to the moduli spaces of colliding particles (Devadoss, Devadoss/Heath/Vipismakul, Devadoss/Fehrman/Heath/Vashist). String interactions generate the moduli spaces of Riemann surfaces (Zwiebach, A First Course in String Theory, pg. 310) with punctures corresponding to particles interacting on a line segment. There is much literature on the relations among compositional inversion/Legendre transformation, Feynman functional/path/gaussian integrals representing partition functions and sums over Feynman diagrams/graphs for point particle interactions (Borcherd pg. 34, Getzler, Manin, Abdesselam, Bergstrom and Brown). Are there analogous arguments directly in terms of sums over moduli spaces for string interactions [as for the beta integral for the Veneziano amplitudes (Zwiebach, pg. 311)] that circumvent the Feynman particle/stable graph interpretations and highlight more directly the connections between compositional inverses/Legendre transforms and the face polynomials of associahedra?