  We find the generating function for the contributions of $n$-cylinder
square-tiled surfaces to the Masur-Veech volume of $\mathcal{H}(2g-2)$. It is a
bivariate generalization of the generating function for the total volumes
obtained by Sauvaget via intersection theory. Our approach is, however, purely
combinatorial. It relies on the study of counting functions for certain
families of metric ribbon graphs. Their top-degree terms are polynomials, whose
(normalized) coefficients are cardinalities of certain families of metric plane
trees. These polynomials are analogues of Kontsevich polynomials that appear as
part of his proof of Witten's conjecture.
