In my answer to MO-Q: Enumerative geometry and nonlinear waves, I outline the relation between the refined face polynomials of the Stasheff polytopes (associahedra) and the partition polynomials for the compositional inverse of a formal power series, noted by Loday in his paper referenced in the answer.

Who was the first to note this relation?

(Cross-posted from HSM.)

Ancillary question: Who was the first to note the relation between the dissections of convex polygons (or, closely related, Cayley trees depicting the repeated action of $g(x)D_x$) and compositional inversion?