Census of Planar Maps: From the One-Matrix Model Solution to a   Combinatorial Proof
  We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees. 
J. Bouttier, P. Di Francesco and E. Guitter (SPHT-Saclay)