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In this paper we prove, in a purely combinatorial way, a structuralquasi-polynomiality property for the Bousquet-M\'elou-Schaeffer numbers.Conjecturally, this property should follow from the Chekhov-Eynard-Orantintopological recursion for these numbers (or, to be more precise, theBouchard-Eynard version of the topological recursion for higher order criticalpoints), which we derive in this paper from the recent result ofAlexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is ageneralization to the case of higher order critical points on the underlyingspectral curve of the existing correspondence between the topological recursionand Givental's theory for cohomological field theories.

Combinatorics of Bousquet-M\'elou-Schaeffer numbers in the light of topological recursion