A bijection for triangulations, quadrangulations, pentagulations, etc

Understand

A $d$-angulation is a planar map with faces of degree $d$.

• We present for each integer $d\geq 3$ a bijection between the class of $d$-angulations of girth $d$ (i.e., with no cycle of length less than $d$) and a class of decorated plane trees.
• Each of the bijections is obtained by specializing a "master bijection" which extends an earlier construction of the first author.
• Our construction unifies known bijections by Fusy, Poulalhon and Schaeffer for triangulations ($d=3$) and by Schaeffer for quadrangulations ($d=4$).