The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size n is CnCn+1 where Cn = 1 n+1 2n n is the n th Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.

A d-angulation is a planar map with faces of degree d. We present for each integer d ≥ 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). For d ≥ 5, both the bijections and the enumerative results are new.We also extend our bijections so as to enumerate p-gonal d-angulations (d-angulations with a simple boundary of length p) of girth d. We thereby recover bijectively the results of Brown for simple p-gonal triangulations and simple 2p-gonal quadrangulations and establish new results for d ≥ 5.A key ingredient in our proofs is a class of orientations characterizing dangulations of girth d. Earlier results by Schnyder and by De Fraysseix and Ossona de Mendez showed that simple triangulations and simple quadrangulations are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a d-angulation has girth d if and only if the graph obtained by duplicating each edge d − 2 times admits an orientation having indegree d at each inner vertex. *

The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size n as the relation of being above. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a previous article, the second author defined a bijection Φ between pairs of non-crossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection Φ. Then, we study the restriction of Φ to Tamari and Kreweras intervals. We prove that Φ induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that Φ induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, Φ induces a bijection between Kreweras intervals and stack triangulations which are known to be in bijection with ternary trees.