. Remark, Alternatively, we could have also used Corollary 12.6.2 with constant order map

. Example, The decision tree associated to the graph from Figure 11

]. O. Ang03 and . Angel, Growth and percolation on the uniform infinite planar triangulation, Geom. Funct. Anal, vol.13, issue.5, pp.935-974, 2003.

D. [. Albenque, . Poulalhon-]-m, I. A. Abramowitz, . J. Stegunbax82-]-r, and . Baxter, Generic method for bijections between blossoming trees and planar maps Handbook of mathematical functions with formulas , graphs, and mathematical tables Exactly solved models in statistical mechanics, National Bureau of Standards Applied Mathematics Series, vol.55, issue.25, pp.149-194, 1964.

O. Bernardi and M. Bousquet-mélou, Counting colored planar maps : differential equations, En préparation, p.77

S. [. Bell, K. A. Burris, . A. Yeatsbc86-]-e, E. R. Bender, . A. Canfieldbc94-]-e et al., Characteristic points of recursive systemsResearch Paper 121 The asymptotic number of rooted maps on a surface The number of degree-restricted rooted maps on the sphere, Electron. J. Combin. J. Combin. Theory Ser. A SIAM J. Discrete Math, vol.17, issue.22, pp.150244-257, 1986.

J. Bouttier, P. D. Francesco, and E. Guitter, Critical and tricritical hard objects on bicolourable random lattices: exact solutions, Journal of Physics A: Mathematical and General, vol.35, issue.17, pp.3821-3854, 2002.
DOI : 10.1088/0305-4470/35/17/302

J. Bouttier, P. D. Francesco, and E. Guitter, Planar maps as labeled mobiles, Electron. J. Combin, vol.11, issue.27, p.37, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00586658

J. Bouttier, P. D. Francesco, and E. Guitter, Combinatorics of bicubic maps with hard particles, Journal of Physics A: Mathematical and General, vol.38, issue.21, pp.4529-4559, 2005.
DOI : 10.1088/0305-4470/38/21/002

URL : https://hal.archives-ouvertes.fr/hal-00586657

J. Bouttier, P. D. Francesco, E. Guitterben74-]-e, and . Bender, Blocked edges on Eulerian maps and mobiles : application to spanning trees, hard particles and the Ising model Asymptotic methods in enumeration, J. Phys. A SIAM Rev, vol.21, issue.16, pp.407411-7440, 1974.

]. D. Ber90, . Bertrandber08a-]-o, and . Bernardi, Extensions de D-modules et groupes de Galois différentiels In p-adic analysis (Trento, 1989), volume 1454 of Lecture Notes in Math A characterization of the Tutte polynomial via combinatorial embedding, Annals of Combinatorics, vol.12, issue.176, pp.125-141, 0200.

O. Bernardi, Tutte polynomial, subgraphs, orientations and sandpile model : new connections via embeddings, Electron. J. Combin, vol.15, issue.53, p.27, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00117268

]. O. Ber12, J. Bernardi, P. D. Bouttier, E. Francesco, and . Guitter, An analogue of the Harer-Zagier formula for unicellular maps on general surfaces Census of planar maps : from the one-matrix model solution to a combinatorial proof, Adv. in Appl. Math. Nuclear Physics B, vol.48, issue.21, pp.164-180, 2002.

C. Banderier, P. Flajolet, G. Schaeffer, M. Soriabg12, ]. J. Bouttier et al., Random maps, coalescing saddles, singularity analysis, and Airy phenomena Planar maps and continued fractions, Random Structures Algorithms Communications in Mathematical Physics, vol.19, issue.309, pp.3-4194, 2001.
DOI : 10.1002/rsa.10021

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.3801

E. Brézin, C. Itzykson, G. Parisi, J. B. Zuberbk87-]-d, V. A. Boulatov et al., Planar diagrams, Communications in Mathematical Physics, vol.16, issue.1, pp.35-513, 1978.
DOI : 10.1007/BF01614153

]. A. Blvs-+-93, M. L. Björner, B. Vergnas, N. Sturmfels, G. White et al., Oriented matroids, volume 46 of Encyclopedia of Mathematics and its Applications, p.26, 1993.

. Bmc-]-m, J. Bousquet-mélou, and . Courtiel, Spanning forests in regular planar maps. En préparation, p.46

M. Bousquet-mélou and J. Courtiel, Spanning forests in regular planar maps, DMTCS Proceedings, pp.241-252
DOI : 10.1016/j.jcta.2015.04.002

A. [. Bousquet-mélou and . Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, Journal of Combinatorial Theory, Series B, vol.96, issue.5, pp.623-672, 2006.
DOI : 10.1016/j.jctb.2005.12.003

M. Bousquet-mélou and G. Schaeffer, The degree distribution in bipartite planar maps : applications to the Ising model ArXiv math, 2002.

M. Bousquet-mélou and G. Schaeffer, The degree distribution of bipartite planar maps : applications to the Ising model, Formal Power Series and Algebraic Combinatorics, pp.312-323, 2003.

J. [. Brylawski and . Oxley, The Tutte Polynomial and Its Applications, Matroid applications, pp.123-225
DOI : 10.1017/CBO9780511662041.007

P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality, Physical Review A, vol.38, issue.1, pp.364-374, 1988.
DOI : 10.1103/PhysRevA.38.364

]. W. Büh92 and . Bühring, Generalized hypergeometric functions at unit argument, Proc

]. F. Car21 and . Carlson, Über Potenzreihen mit ganzzahligen Koeffizienten, Math. Z, vol.9, issue.12, pp.1-13, 1921.

[. Cori and Y. L. Borgne, The sand-pile model and Tutte polynomials, Advances in Applied Mathematics, vol.30, issue.1-2, pp.44-52, 2003.
DOI : 10.1016/S0196-8858(02)00524-9

URL : https://hal.archives-ouvertes.fr/hal-00307567

[. Cori and A. Machi, Maps, hypermaps and their automorphisms : a survey . I, II, III, Exposition. Math, vol.10, issue.5, pp.403-467, 1992.

G. Chapuy, M. Marcus, and G. Schaeffer, A Bijection for Rooted Maps on Orientable Surfaces, SIAM Journal on Discrete Mathematics, vol.23, issue.3, pp.1587-1611, 2009.
DOI : 10.1137/080720097

URL : https://hal.archives-ouvertes.fr/hal-00713482

R. Cori, Un code pour les graphes planaires et ses applications With an English abstract, Astérisque, No. 27. 171 [Cra69] H. H. Crapo. The Tutte polynomial, Aequationes Math, vol.3, issue.169, pp.211-229, 1969.

[. Cori and B. Vauquelin, Planar maps are well labeled trees. Canad, J. Math, vol.33, issue.21, pp.1023-1042, 1981.
DOI : 10.4153/cjm-1981-078-2

P. , D. Francesco, B. Eynard, and E. Guitter, Coloring random triangulations, Nuclear Phys. B, vol.516, issue.25, pp.543-587, 1998.

]. D. Dha90 and . Dhar, Self-organized critical state of sandpile automaton models Duplantier and I. Kostov. Conformal spectra of polymers on a random surface, Phys. Rev. Lett. Phys. Rev. Lett, vol.64, issue.1413, pp.1613-1616, 1988.

. [. Dvoretzky, . Th, and . Motzkin, A problem of arrangements, Duke Mathematical Journal, vol.14, issue.2, pp.305-313, 1947.
DOI : 10.1215/S0012-7094-47-01423-3

M. Drmota, Systems of functional equations. Random Structures Algorithms, pp.103-124, 1997.
URL : https://hal.archives-ouvertes.fr/hal-01197227

M. Drmota, Combinatorics and asymptotics on trees, Cubo Journal, vol.6, p.116, 2004.

G. [. Eynard and . Bonnet, The Potts-q random matrix model: loop equations, critical exponents, and rational case, Physics Letters B, vol.463, issue.2-4, pp.273-279, 1972.
DOI : 10.1016/S0370-2693(99)00925-9

R. [. Flajolet, . Sedgewickgs96-]-i, B. Gessel, and . Sagan, Analytic combinatorics Weight enumeration and the geometry of linear codes The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions, Studies in Appl. Math. The Electronic Journal of Combinatorics, vol.115, issue.178, pp.152119-128, 1976.

L. [. Gordon and . Traldi, Generalized activities and the tutte polynomial, Discrete Mathematics, vol.85, issue.2, pp.167-176, 0198.
DOI : 10.1016/0012-365X(90)90019-E

]. V. Kaz86 and . Kazakov, Ising model on a dynamical planar random lattice : exact solution, Phys. Lett. A, vol.119, issue.3, pp.140-144, 1986.

]. S. Lan02, . Lang, and . Algebra, Uniqueness and universality of the Brownian map, Ann. Probab, vol.70, issue.414 21, pp.2880-2960, 2002.

]. L. Lip88 and . Lipshitz, The diagonal of a D-finite power series is D-finite, J. Algebra, vol.113, issue.17, pp.373-378, 1988.

]. L. Lip89, . Lipshitzmie13-]-g, and . Miermont, D-finite power series The Brownian map is the scaling limit of uniform random plane quadrangulations, ML97] C. Merino López. Chip firing and the Tutte polynomial, pp.353-373, 1989.

]. R. Mul67, . M. Mullinor83-]-a, L. B. Odlyzko, and . Richmond, On the enumeration of tree-rooted maps A differential equation arising in chromatic sum theory, Proceedings of the fourteenth Southeastern conference on combinatorics, graph theory and computing, pp.174-183, 1967.

D. [. Oxley, O. Welsh-waterloo, . Sa92-]-d, A. Stauffer, and . Aharony, The Tutte polynomial and percolation Introduction to percolation theory. Taylor and Francis London Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees, Graph theory and related topics (Proc. Conf. Conjugaison d'arbres et cartes combinatoires aléatoiresSS11] O. Schramm and S. Smirnov. On the scaling limits of planar percolation, pp.329-339, 1977.

. Ann and . Probab, With an appendix by Christophe Garban, pp.1768-1814, 2011.

]. R. Sta99, . P. Stanleysta12-]-r, and . Stanley, Enumerative combinatorics of Cambridge Studies in Advanced Mathematics, Enumerative combinatorics. Cambridge Studies in Advanced Mathematics, vol.2, issue.13, pp.118-133, 1999.

P. [. Salvy, M. B. Zimmermann, . T. Thistlethwaitetut54-]-w, . T. Tuttetut62-]-w, and . Tutte, A spanning tree expansion of the Jones polynomial A contribution on the theory of chromatic polynomial A census of planar triangulations. Canad A census of planar maps Asymptotic expansions for ordinary differential equations, ACM Transactions on Mathematical Software Topology Canadian Journal of Mathematics J. Math. Canad. J. Math. Pure and Applied Mathematics Math. Soc, vol.20, issue.38, pp.163-177297, 1932.

]. F. Wu82 and . Wu, The Potts model, Reviews of Modern Physics, vol.54, issue.1, pp.235-268, 1925.

. [. Zinn-justin, The dilute Potts model on random surfaces, Journal of Statistical Physics, vol.98, issue.1/2, pp.245-264, 2000.
DOI : 10.1023/A:1018626906256

. Note, Les numéros en fin de chaque référence correspondent aux pages du mémoire dans lesquelles cette référence a été citée