, Le cas particulier suivant est fréquent, notamment pour la propriété de martingale

, une des tribus d'une filtration (F n ) n ?0. Ceci se produit lorsque l'on considère un processus à temps discret (X n ) n ?0 , et F n est la tribu

, Comme L 2 est dense dans L 1 , la notion s'étend aux fonctions intégrables. Et la caractérisation suivante est plus utile en pratique que la définition

, Soit X ? L 1 (?, A, P) et soit B une sous-tribu de A. Alors E(X |B) est l'unique variable aléatoire telle que (i) E(X |B) est B-mesurable

, Propriétés de l'espérance conditionnelle). (i) Linéarité : ?a, Proposition C, vol.20

. Si-c-est-une-tribu-et-si-c-?-b,

. Si-x-est-intégrable and . Dans, , vol.1

, Cay : arbres de Cayley.-DST(Y ) : arbre digital de recherche associé à une suite Y de mots.-M : arbres m-aires (non marqués).-P : arbres planaires.-Pó : arbres non planaires, ou arbres de Pólya.-P (2) : arbres de Pólya binaires

, Si T est un ensemble d'arbres, la notation T n désigne le sous-ensemble des arbres de taille n

, nous utilisons la convention suivante :-? pour le cas général d'un noeud dans un arbre, et pour les noeuds internes d'un arbre binaire complété.-pour une feuille d'un arbre général, ou d'un arbre complété

E. , 3 Paramètres d'un arbre ?-Taille de l'arbre : |?|.-Hauteur : h(?).-Niveau de saturation : s(?)

, Longueur de cheminement : lc(?) ; longueurs de cheminement interne lci(?) et externe lce(?),-Profondeur d

, Profondeur d'insertion d'une clé dans un arbre de recherche ? n : D n+1 .-Nombre de feuilles qui se trouvent au niveau p : U p (?).-Polynôme de niveaux : W ? (z)

, ensemble de symboles lié à une famille simple d'arbres

, E.2.4 Arbres et sous-arbres associés à un arbre donné ?-? arbre binaire : sous-arbres gauche et droit ? (g) et ? (d)

. Si-u-est-un-noeud-de-?, alors ? u est le sous-arbre de ? dont la racine est u.-? arbre planaire non réduit à une feuille : ? i est le i-ième sous-arbre de la racine

, ? arbre marqué : ?(?) est la forme de ? ; C(?) est l'arbre des rangs, obtenu par marquage canonique de ?

E. , Constructions combinatoires-Seq : suite-de-Seq >0 : suite-non-vide-de-Set

A. Si, A * est la suite des mots finis sur A ; ? ? A * est le mot vide

L. |y-=-y) and L. |y,

, B) pour deux événements A et B : signifie P(A ? B) ou encore P(A ? B). ? x : mesure de Dirac en x : mesure de masse 1 sur l'ensemble {x}

, A : fonction indicatrice de l'ensemble A : 1 1 A (x) = 1 quand x ? A et 1 1 A (x) = 0 quand x A. Conséquence : 1 1 A (x) = ? x (A)

X. +-est-la-partie-positive-de-la-variable-aléatoire and X. ,

, Loi Ord sur l'ensemble des arbres binaires de recherche, dite aussi modèle des permutations uniformes : lorsque les clés insérées sont indépendantes et de même loi

, Loi Ord d sur les arbres quadrants de recherche, quand les clés sont prises dans un ensemble qui est un produit cartésien de d ensembles

, On-line encyclopaedia of integer sequences

G. Adel'son-vel'skii and E. Landis, An algorithm for organisation of information, Traduction anglaise : Soviet Math. Dokl, vol.146, pp.1259-1263, 1962.

A. V. Aho and N. J. Sloane, Some doubly exponential sequences. Fibonacci Quarterly, pp.429-437, 1970.

D. Aldous, The continuum random tree. i, Annals of Probab, vol.19, issue.1, pp.1-28, 1991.

D. Aldous, The continuum random tree. ii : an overview, Stochastic Analysis, pp.23-70, 1991.

D. Aldous, The continuum random tree. iii, Annals of Probab, vol.21, issue.1, pp.248-289, 1993.

A. Apostolico, The myriad virtues of subword trees, Combinatorial Algorithms on Words, vol.12, pp.85-96, 1985.

A. Apostolico, M. Crochemore, M. Farach-colton, Z. Galil, and S. Muthukrishnan, 40 years of suffix trees, Commun. ACM, vol.59, issue.4, pp.66-73, 2016.

C. R. Aragon and R. G. Seidel, Randomized search trees, Foundations of Computer Science, pp.540-545, 1989.

C. R. Aragon and R. G. Seidel, Randomized search trees, Algorithmica, vol.16, issue.4, pp.464-497, 1996.

S. Asmussen and H. Hering, Branching processes, 1983.

K. Athreya, On a characteristic property of Pólya's urn, Studia Sci. Math. Hungar, vol.4, pp.31-35, 1969.

K. Athreya and P. Ney, Branching processes, 1972.

M. Aumüller and M. Dietzfelbinger, Optimal partitioning for dual pivot quicksort, Automata, Languages and Programming, vol.7965, pp.33-44, 2013.

C. Banderier, M. Bousquet-mélou, A. Denise, P. Flajolet, D. Gardy et al., Generating functions for generating trees, Discrete Mathematics, vol.246, issue.1-3, pp.29-55, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00003258

P. Barbe, M. Ledoux, and . Probabilité, Collection : enseignement SUP-Maths, 2007.

E. Barcucci, A. D. Lungo, and E. Pergola, Random generation of trees and other combinatorial objects, Theoretical Computer Science, vol.218, issue.2, pp.219-232, 1999.

E. Barcucci, A. Del-lungo, E. Pergola, and R. Pinzani, A methodology for plane tree enumeration, Discrete Mathematics, vol.180, issue.1-3, pp.45-64, 1998.

T. Bell, I. H. Witten, and J. G. Cleary, Modelling for text compression, ACM Computing Surveys, vol.21, issue.4, pp.557-591, 1989.

J. L. Bentley and R. Sedgewick, Fast algorithms for sorting and searching strings, Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '97, pp.360-369, 1997.

B. Bercu and D. Chafaï, Modélisation stochastique et simulation : cours et applications, Série « Mathématiques appliquées pour le Master / SMAI, 2007.

J. Berstel, D. Perrin, and C. Reutenauer, Codes and Automata (Encyclopedia of Mathematics and Its Applications), 2009.

J. Bertoin, Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00103015

P. Biane, J. Pitman, and M. Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and brownian excursions, Bull. Amer. Math. Soc, vol.38, pp.435-465, 2001.

J. Biggins, Chernoff's theorem in the branching random walk, Journal of Applied Probability, vol.14, pp.630-636, 1977.

J. D. Biggins, Martingale convergence in the branching random walk, Advances in Applied Probability, vol.10, pp.62-84, 1978.

J. D. Biggins, Uniform convergence of martingales in the branching random walk. The Annals of Probability, vol.20, pp.137-151, 1992.

J. D. Biggins, How fast does a general branching random walk spread ? Classical and modern branching processes, Math. Appl, vol.84, pp.19-39, 1997.

P. Billingsley, Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 1995.

D. Blackwell and D. Kendall, The Martin boundary for Pólya's urn and an application to stochastic population growth, Journal of Applied Probability, vol.1, issue.2, pp.284-296, 1964.

B. Bollobas and I. Simon, Repeated random insertion into a priority queue, Journal of Algorithms, vol.6, pp.466-477, 1985.

N. Broutin, L. Devroye, E. Mcleish, and M. De-la-salle, The height of increasing trees, Random Structures & Algorithms, vol.32, pp.494-518, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01219519

J. Capetanakis, Tree algorithms for packet broadcast channels, IEEE Transactions on Information Theory, IT, vol.25, pp.505-515, 1979.

E. Cesaratto and B. Vallée, Gaussian distribution of trie depth for strongly tame sources, Combinatorics, Probability & Computing, vol.24, issue.1, pp.54-103, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01153052

P. Chassaing and P. Flajolet, Hachage, arbres, chemins et graphes, Gazette des Mathématiciens, vol.95, pp.29-49, 2003.

P. Chassaing and J. Marckert, Parking functions, empirical processes and the width of rooted labeled trees, The Electronic Journal of Combinatorics, vol.8, p.14, 2001.

B. Chauvin, M. Drmota, and J. Jabbour-hattab, The profile of binary search trees, Annals of Applied Probability, vol.11, pp.1042-1062, 2001.

B. Chauvin, P. Flajolet, D. Gardy, and B. Gittenberger, And/or trees revisited, Combinatorics, Probability & Computing, vol.13, issue.4-5, pp.475-497, 2004.

B. Chauvin, D. Gardy, and C. Mailler, A sprouting tree model for random boolean functions, Random Structures & Algorithms, vol.47, issue.4, pp.635-662, 2015.
URL : https://hal.archives-ouvertes.fr/hal-02063524

B. Chauvin, D. Gardy, N. Pouyanne, and D. , Ton That. B-urns. Latin American Journal Of Probability And Mathematical Statistics, vol.13, pp.605-634, 2016.

B. Chauvin, T. Klein, J. Marckert, and A. Rouault, Martingales and profile of binary search trees, Electronic Journal in Probability, vol.10, pp.420-435, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00138808

B. Chauvin, C. Mailler, and N. Pouyanne, Smoothing equations for large Pólya urns, Journal of Theoretical Probability, vol.28, pp.923-957, 2015.

B. Chauvin and N. Pouyanne, m-ary search trees when m > 26 : a strong asymptotics for the space requirements, Random Structures & Algorithms, vol.24, issue.2, pp.133-154, 2004.

B. Chauvin, N. Pouyanne, and R. Sahnoun, Limit distributions for large Pólya urns, Annals of Applied Probability, vol.21, issue.1, pp.1-32, 2011.

H. Chern, M. Fuchs, and H. Hwang, Phase changes in random point quadtrees, ACM Transactions on Algorithms, vol.3, 2007.

J. Clément, P. Flajolet, and B. Vallée, The analysis of hybrid trie structures, Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.531-539, 1998.

J. Clément, P. Flajolet, and B. Vallée, Dynamical sources in information theory : A general analysis of trie structures, Algorithmica, vol.29, issue.1, pp.307-369, 2001.

J. Clément, T. Nguyen-thi, and B. Vallée, A general framework for the realistic analysis of sorting and searching algorithms, STACS, 30th Annual Symposium on Theoretical Aspects of Computer Science, pp.598-609, 2013.

J. Clément, T. H. Thi, and B. Vallée, Towards a realistic analysis of some popular sorting algorithms, Combinatorics, Probability & Computing, vol.24, issue.1, pp.104-144, 2015.

D. Comer, The uniquitous B-tree, ACM Computing Surveys, vol.11, issue.2, pp.121-137, 1979.

T. Cormen, C. Leiserson, and R. Rivest, , 2002.

T. Cormen, C. Leiserson, R. Rivest, and C. Stein, , 2010.

M. Crochemore, C. Hancart, and T. Lecroq, Algorithms on Strings, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00470109

R. De-la and . Briandais, File searching using variable length keys, Western Joint Computer Conference, IREAIEE-ACM '59 (Western), pp.295-298, 1959.

A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 1998.

L. Devroye, A probabilistic analysis of the height of tries and of the complexity of triesort, Acta Informatica, vol.21, pp.229-237, 1984.

L. Devroye, Branching processes in the analysis of the height of trees, Acta Informatica, vol.24, pp.277-298, 1987.

L. Devroye, Applications of the theory of records in the study of random trees, Acta Informatica, vol.26, pp.123-130, 1988.

L. Devroye, A study of trie-like structures under the density model, Annals of Applied Probability, vol.2, issue.2, pp.402-434, 1992.

L. Devroye, Branching Processes and Their Applications in the Analysis of Tree Structures and Tree Algorithms, Probabilistic Methods for Algorithmic Discrete Mathematics, vol.16, pp.249-314, 1998.

L. Devroye, An analysis of random LC tries, Random Structures & Algorithms, vol.15, pp.359-375, 2001.

L. Devroye and L. Laforest, An analysis of random d-dimensional quadtrees, SIAM Journal of Computing, vol.19, issue.5, pp.821-832, 1990.

L. Devroye and B. Reed, On the variance of the height of random binary search trees, SIAM Journal of Computing, vol.24, pp.1157-1162, 1995.

E. Doberkat, Some observations on the average behavior of heapsort (preliminary report), Int. Conf. on the Foundations of Computer Science (FOCS), pp.229-237, 1980.

E. Doberkat, Deleting the root of a heap, Acta Informatica, vol.17, pp.245-265, 1982.

E. Doberkat, An average case analysis of Floyd's algorithm to construct heaps, Information and Control, vol.61, issue.2, pp.114-131, 1984.

J. Doyle and R. Rivest, Linear expected time of a simple union-find algorithm, Inform. Proc. Lett, vol.5, pp.146-148, 1976.

M. Drmota, The variance of the height of binary search trees, Theoretical Computer Science, vol.270, pp.913-919, 2002.

M. Drmota, Random Trees. An Interplay between Combinatorics and Probabilities, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00001281

M. Drmota and B. Gittenberger, On the profile of random trees, Random Structures & Algorithms, vol.10, pp.421-451, 1997.

M. Drmota and B. Gittenberger, The width of Galton-Watson trees conditioned by the size, Discr. Math. Theor. Comput. Sci, vol.6, issue.2, pp.387-400, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00959015

R. Dudley, Real Analysis and Probability, 2002.

T. Duquesne and J. Gall, Random trees, Levy processes and spatial branching processes, Asterisque, vol.281, 2002.

M. Durand and P. Flajolet, Loglog counting of large cardinalities, ESA03, 11th Annual European Symposium on Algorithms, Engineering and Applications Track, pp.605-617, 2003.

R. Durrett, Probability. Theory and Examples. Cambridge, 2010.

R. Fagin, J. Nievergelt, N. Pippenger, and H. R. Strong, Extendible hashing-a fast access method for dynamic files, ACM Transactions on Database Systems, vol.4, issue.3, pp.315-344, 1979.

J. Fill and S. Janson, Smoothness and decay properties of the limiting quicksort density function, Mathematics and Computer Science, Trends in Mathematics, pp.53-64, 2000.

J. Fill, H. Mahmoud, and W. Szpankowski, On the distribution for the duration of a randomized leader election algorithm, Annals of Applied Probability, vol.6, issue.4, pp.1260-1283, 1996.

J. A. Fill and S. Janson, The number of bit comparisons used by Quicksort : an average-case analysis, Electron. J. Probab, vol.17, issue.43, pp.1-22, 2012.

P. Flajolet, On the performance evaluation of extendible hashing and trie searching, Acta Informatica, vol.20, pp.345-369, 1983.

P. Flajolet, Approximate counting : a detailed analysis, Bit, vol.25, pp.113-134, 1985.

P. Flajolet, On adaptive sampling. Computing, vol.43, pp.391-400, 1990.
URL : https://hal.archives-ouvertes.fr/inria-00075533

P. Flajolet, Counting by coin tossing, Proceedings of Asian'04, number 3321 in Lecture Notes in Computer Science, pp.1-12

. Springer, , 2004.

P. Flajolet, The ubiquitous digital tree, STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, vol.3884, pp.1-22, 2006.

P. Flajolet, P. Dumas, and V. Puyhaubert, Some exactly solvable models of urn process theory, DMTCS Proceedings, pp.59-118, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01184710

P. Flajolet, E. Fusy, O. Gandouet, and F. Meunier, Hyperloglog : the analysis of a near-optimal cardinality estimation algorithm, DMTCS Proceedings, pp.127-146, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00406166

P. Flajolet, J. Gabarró, and H. Pekari, Analytic urns, Annals of Probability, vol.33, issue.3, pp.1200-1233, 2005.
DOI : 10.1214/009117905000000026
URL : https://doi.org/10.1214/009117905000000026

P. Flajolet, Z. Gao, A. Odlyzko, and B. Richmond, The distribution of heights of binary trees and other simple trees, Combinatorics, Probability & Computing, vol.2, pp.145-156, 1993.
URL : https://hal.archives-ouvertes.fr/inria-00076989

P. Flajolet, G. Gonnet, C. Puech, and J. Robson, Analytic variations on quadtrees, Algorithmica, vol.10, issue.6, pp.473-500, 1993.
DOI : 10.1007/bf01891833

P. Flajolet, X. Gourdon, and P. Dumas, Mellin transforms and asymptotics : Harmonic sums, Theoretical Computer Science, vol.144, issue.1-2, pp.3-58, 1995.
DOI : 10.1016/0304-3975(95)00002-e
URL : https://hal.archives-ouvertes.fr/inria-00074307

P. Flajolet and M. Hoshi, Page usage in a quadtree index, BIT, vol.32, pp.384-402, 1992.

P. Flajolet and P. Jacquet, Analytic models for tree communication protocols, Flow Control of Congested Networks, pp.223-234, 1987.
DOI : 10.1007/978-3-642-86726-2_13
URL : https://hal.archives-ouvertes.fr/inria-00075905

P. Flajolet, G. Labelle, L. Laforest, and B. Salvy, Hypergeometrics and the cost structure of quadtrees, Random Structures & Algorithms, vol.7, issue.2, pp.117-144, 1995.
URL : https://hal.archives-ouvertes.fr/inria-00074422

P. Flajolet and T. Lafforgue, Search costs in quadtrees and singularity perturbation asymptotics, Discrete & Computational Geometry, vol.12, issue.4, pp.151-175, 1994.
DOI : 10.1007/bf02574372
URL : https://hal.archives-ouvertes.fr/inria-00074811

P. Flajolet and G. Martin, Probabilistic counting algorithms for data base applications, Journal of Computer and System Sciences, vol.31, issue.2, pp.182-209, 1985.
DOI : 10.1016/0022-0000(85)90041-8
URL : https://hal.archives-ouvertes.fr/inria-00076244

P. Flajolet and A. Odlyzko, The average height of binary trees and other simple trees, Journal of Computer and System Sciences, vol.25, pp.171-213, 1982.
URL : https://hal.archives-ouvertes.fr/inria-00076505

P. Flajolet and A. Odlyzko, Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations, Mathematical Proceedings of the Cambridge Philosophical Society, vol.96, issue.2, 1984.
URL : https://hal.archives-ouvertes.fr/inria-00076276

P. Flajolet and A. M. Odlyzko, Singularity analysis of generating functions, SIAM Journal on Discrete Mathematics, vol.3, issue.2, pp.216-240, 1990.
DOI : 10.1017/cbo9780511801655.008
URL : https://hal.archives-ouvertes.fr/inria-00075725

P. Flajolet, M. Régnier, and D. Sotteau, Algebraic methods for trie statistics, Analysis and Design of Algorithms for Combinatorial Problems, vol.25, pp.145-188, 1985.
DOI : 10.1016/s0304-0208(08)73107-4
URL : https://hal.archives-ouvertes.fr/inria-00076259

P. Flajolet and B. Richmond, Generalized digital trees and their difference-differential equations, Random Structures & Algorithms, vol.3, issue.3, pp.305-320, 1992.
DOI : 10.1002/rsa.3240030309
URL : https://hal.archives-ouvertes.fr/inria-00075137

P. Flajolet, M. Roux, and B. Vallée, Digital trees and memoryless sources : from arithmetics to analysis, International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms, pp.233-260
URL : https://hal.archives-ouvertes.fr/hal-01083405

P. Flajolet and R. Sedgewick, Digital search trees revisited, SIAM J. Comput, vol.15, issue.3, pp.748-767, 1986.
DOI : 10.1137/0215054
URL : https://hal.archives-ouvertes.fr/inria-00076246

P. Flajolet and R. Sedgewick, Mellin transforms and asymptotics : Finite differences and Rice's integrals, Theoretical Computer Science, vol.144, issue.1&2, pp.101-124, 1995.
DOI : 10.1016/0304-3975(94)00281-m
URL : https://doi.org/10.1016/0304-3975(94)00281-m

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009.
DOI : 10.1017/cbo9780511801655
URL : https://hal.archives-ouvertes.fr/inria-00072739

P. Flajolet and J. Steyaert, A complexity calculus for classes of recursive search programs over tree structures, Foundations of Computer Science, 1981. SFCS'81. 22nd Annual Symposium on, pp.386-393, 1981.

P. Flajolet and J. Steyaert, A branching process arising in dynamic hashing, trie searching and polynomial factorization, Automata, Languages and Programming, vol.140, pp.239-251, 1982.
DOI : 10.1007/bfb0012773

R. Floyd, Algorithm 24. Treesort 3, Communications of the ACM, vol.7, issue.12, 1964.

H. Fournier, D. Gardy, A. Genitrini, and B. Gittenberger, The fraction of large random trees representing a given boolean function in implicational logic, Random Structures & Algorithms, vol.20, issue.7, pp.875-887, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00551234

H. Fournier, D. Gardy, A. Genitrini, and M. Zaionc, Classical and intuitionistic logic are asymptotically identical, Computer Science Logic, 21st International Workshop, pp.177-193, 2007.
DOI : 10.1007/978-3-540-74915-8_16
URL : https://hal.archives-ouvertes.fr/hal-00530534

E. Fredkin, Trie memory, Communications of the ACM, vol.3, issue.9, pp.490-499, 1960.
DOI : 10.1145/367390.367400

A. Frieze, On the random construction of heaps, Information Processing Letters, vol.27, pp.103-109, 1988.

C. Froidevaux, M. Gaudel, and M. Soria, Types de données et Algorithmes. McGraw Hill, 1993.

M. Fuchs, H. Hwang, and R. Neininger, Profiles of random trees : Limit theorems for random recursive trees and binary search trees, Algorithmica, vol.46, pp.367-407, 2006.
DOI : 10.1007/s00453-006-0109-5
URL : http://algo.stat.sinica.edu.tw/hk/files/2005_08/pdf/Profiles_of_random_trees_Limit_theorems.pdf

M. Fuchs, H. Hwang, and V. Zacharovas, An analytic approach to the asymptotic variance of trie statistics and related structures, Theoretical Computer Science, vol.527, pp.1-36, 2014.

Z. Galil and G. N. Italiano, Data structures and algorithms for disjoint set union problems, ACM Computing Surveys, vol.2, issue.3, 1991.
DOI : 10.1145/116873.116878
URL : https://academiccommons.columbia.edu/doi/10.7916/D8FB5B0J/download

D. Gardy, Random boolean expressions, Colloquium on Computational Logic and Applications, pp.1-36, 2005.
URL : https://hal.archives-ouvertes.fr/hal-01183339

O. Garet and A. Kurtzmann, De l'intégration aux probabilités. Ellipses, 2011.

A. Genitrini, B. Gittenberger, V. Kraus, and C. Mailler, Associative and commutative tree representations for boolean functions, Theoretical Computer Science, vol.570, pp.70-101, 2015.
DOI : 10.1016/j.tcs.2014.12.025
URL : https://hal.archives-ouvertes.fr/hal-01122776

A. Genitrini and J. Kozik, In the full propositional logic, 5/8 of classical tautologies are intuitionistically valid, Annals of Pure and Applied Logic, vol.163, issue.7, pp.875-887, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00618303

A. Genitrini and C. Mailler, Generalised and quotient models for random and/or trees and application to satisfiability, Algorithmica, vol.76, issue.4, pp.1106-1138, 2016.
DOI : 10.1007/s00453-016-0113-3
URL : https://hal.archives-ouvertes.fr/hal-01329255

R. Gouet, Martingale functional central limit theorems for a generalized Pólya urn, Annals of Probability, vol.21, issue.3, pp.1624-1639, 1993.
DOI : 10.1214/aop/1176989134
URL : https://doi.org/10.1214/aop/1176989134

R. Gouet, Strong convergence of proportions in a multicolor Pólya urn, Journal of Applied Probability, vol.34, pp.426-435, 1997.

R. L. Graham, D. E. Knuth, and O. Patashnik, Mathématiques concrètes. Intern, 1998.

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Birkhäuser, 1981.

G. Grimmett, Random labelled trees and their branching networks, J. Austral. Math. Soc. (Ser. A), vol.30, pp.229-237, 1980.
DOI : 10.1017/s1446788700016517
URL : https://www.cambridge.org/core/services/aop-cambridge-core/content/view/71BF181E26452018E616253733D50DDE/S1446788700016517a.pdf/div-class-title-random-labelled-trees-and-their-branching-networks-div.pdf

G. Grimmett and D. Stirzaker, Probability and Random Processes, 1982.

D. Gusfield, Algorithms on Strings, Trees, and Sequences : Computer Science and Computational Biology, 1997.

R. Hayward and C. Mcdiarmid, Average-case analysis of heap building by repeated insertion, Journal of Algorithms, vol.12, issue.1, pp.126-153, 1991.
DOI : 10.1016/0196-6774(91)90027-v
URL : http://webdocs.cs.ualberta.ca/~hayward/papers/heap.pdf

P. Hennequin, Combinatorial analysis of quicksort algorithm, vol.23, pp.317-333, 1989.

P. Hennequin, Analyse en moyenne d'algorithmes : Tri rapide et arbres de recherche, vol.170, 1991.

E. Hille, Analytic Function Theory, 1962.

C. Hoare, Quicksort. Computer Journal, pp.10-15, 1962.

M. Hofri, Probabilistic Analysis of Algorithms, 1987.

M. Hofri, Analysis of Algorithms. Computational Methods and Mathematical Tools, 1995.
URL : https://hal.archives-ouvertes.fr/inria-00076123

Y. Hu and Z. Shi, Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees, Annals of Probability, vol.37, issue.2, pp.742-789, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00133596

K. Hun and B. Vallée, Typical depth of a digital search tree built on a general source, 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics, pp.1-15, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01087072

H. Hwang, Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques, 1994.

H. Hwang, On convergence rates in the central limit theorems for combinatorial structures, European Journal of Combinatorics, vol.19, issue.3, pp.329-343, 1998.

H. Hwang and J. Steyaert, On the number of heaps and the cost of heap construction, Second Colloquium on Mathematics and Computer Science : Algorithms, Trees and Combinatorics, pp.295-310, 2002.

J. Jabbour-hattab, Martingales and large deviations for binary search trees. Random Structures & Algorithms, vol.19, pp.112-127, 2001.

J. Jabbour-hattab, Une approche probabiliste du profil des arbres binaires de recherche, 2001.

P. Jacquet and M. Régnier, Trie partitioning process : Limiting distributions, Proceedings of the 11th Colloquium on Trees in Algebra and Programming, vol.214, pp.196-210, 1986.

P. Jacquet and M. Régnier, New results on the size of tries, IEEE Transactions on Information Theory, vol.35, issue.1, pp.203-205, 1989.

P. Jacquet and W. Szpankowski, Analysis of digital tries with markovian dependency, IEEE Transactions on Information Theory, vol.37, issue.5, pp.1470-1475, 1991.

P. Jacquet and W. Szpankowski, Autocorrelation on words and its applications : analysis of suffix trees by string-ruler approach, Journal of Combinatorial Theory (A), vol.66, issue.2, pp.237-269, 1994.
URL : https://hal.archives-ouvertes.fr/inria-00075453

P. Jacquet and W. Szpankowski, Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees, Theoretical Computer Science, vol.144, issue.1-2, pp.161-197, 1995.

P. Jacquet and W. Szpankowski, Analytic pattern matching : from DNA to Twitter, 2015.

S. Janson, Functional limit theorem for multitype branching processes and generalized Pólya urns, Stochastic Processes and their Applications, vol.110, pp.177-245, 2004.

S. Janson, Conditioned Galton-Watson trees do not grow, Fourth Colloquium on Mathematics and Computer Science Algorithms : Trees, Combinatorics and Probabilities, pp.331-334, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01184685

S. Janson, Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation, Probability Surveys, vol.9, pp.103-252, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01197228

S. Janson and W. Szpankowski, Analysis of an asymmetric leader election algorithm, Electronic Journal of Combinatorics, vol.9, 1997.
URL : https://hal.archives-ouvertes.fr/inria-00073602

N. Johnson and S. Kotz, Urn Models and Their Application, 1977.

D. Kennedy, The distribution of the maximum Brownian excursion, J. Appl. Probab, vol.13, pp.371-376, 1976.

H. Kesten, Subdiffusive behavior of random walk on a random cluster, Ann. Inst. Henri Poincaré, vol.22, pp.425-487, 1986.

L. Khizder, Some combinatorial properties of dyadic trees, USSR Comput. Math. and Math. Phys. (traduction anglaise de Zh. Vychisl. Matem. i Matem. Fiziki, vol.6, issue.2, pp.283-290, 1966.

M. Knape and R. Neininger, Pólya urns via the contraction method, 2013.

D. Knuth, Sorting and Searching, The Art of Computer Programming, vol.3, 1998.

D. Knuth and A. Schonhage, The expected linearity of a simple equivalence algorithm, Theor. Comput. Sci, vol.6, pp.281-315, 1978.

D. E. Knuth, Fundamental Algorithms, vol.1, 1973.

V. Kolchin, Branching processes, random trees, and a generalized scheme of arrangements of particles, Math. Notes, vol.21, pp.386-394, 1977.

V. Kolchin, Random Mappings. Optimization Software Inc. Publications Division, 1986.

S. Kotz, H. Mahmoud, and P. Robert, On generalized Pólya urn models, Statistics & Probability Letters, vol.49, pp.163-173, 2000.

R. Kruse and A. Ryba, Data Structures and Program Design in C++, 2000.

P. Larson, Dynamic hashing, BIT, vol.18, issue.2, pp.184-201, 1978.

H. Lefmann and P. Savický, Some typical properties of large and/or boolean formulas, Random Struct. Algorithms, vol.10, issue.3, pp.337-351, 1997.

W. Litwin, Virtual hashing : A dynamically changing hashing, Fourth International Conference on Very Large Data Bases, pp.517-523, 1978.

G. Louchard, Trie size in a dynamic list structure, Random Structures & Algorithms, vol.5, issue.5, pp.665-702, 1994.

G. Louchard, The asymmetric leader election algorithm : number of survivors near the end of the game, Quaestiones Mathematicae, vol.39, issue.1, pp.83-101, 2016.

G. Louchard, H. Prodinger, and M. Ward, Number of survivors in the presence of a demon, vol.64, pp.101-117, 2012.

W. C. Lynch, More combinatorial properties of certain trees, The Computer Journal, vol.7, issue.4, pp.299-302, 1965.

R. Lyons, A simple path to Biggins' martingale convergence for branching random walk. Classical and Modern Branching Processes, Athreya and Jagers. IMA Volumes in Mathematics and its Applications, vol.84, 1997.

R. Lyons, R. Pemantle, and Y. Peres, Conceptual proofs of l log l criteria for mean behavior of branching processes, Annals of Probability, vol.23, pp.1125-1138, 1995.

H. Mahmoud, Evolution of Random Search Trees, 1992.

H. Mahmoud, Pólya urn models, 2008.

H. Mahmoud and B. Pittel, On the most probable shape of a search tree grown from a random permutation, Siam J. Alg. Disc. Meth, vol.5, issue.1, pp.69-81, 1984.

H. Mahmoud and B. Pittel, Analysis of the space of search trees under the random insertion algorithm, Journal of Algorithms, vol.10, pp.52-75, 1989.

H. Mahmoud and R. T. Smythe, Probabilistic analysis of bucket recursive trees, Theoretical Computer Science, vol.144, pp.221-249, 1995.
DOI : 10.1016/0304-3975(94)00308-6
URL : https://doi.org/10.1016/0304-3975(94)00308-6

P. Major, On the invariance principle for sums of independent identically distributed random variables, Journal of Multivariate Analysis, vol.8, pp.487-517, 1978.

J. Marckert, Limit of random walks and of random trees, 2009.

C. Martínez, M. E. Nebel, and S. Wild, Analysis of branch misses in quicksort, Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2015, pp.114-128, 2015.

C. Martínez, D. Panario, and A. Viola, Adaptive sampling strategies for quickselect, ACM Transactions on Algorithms, vol.6, issue.3, 2010.

C. Martínez, A. Panholzer, and H. Prodinger, On the number of descendants and ascendants in random search trees, Electronic Journal of Combinatorics, vol.5, issue.1, p.20, 1998.

C. Martínez and S. Roura, Randomized binary search trees, Journal of the ACM, vol.45, issue.2, pp.288-323, 1998.

C. Martínez and S. Roura, Optimal sampling strategies in quicksort and quickselect, SIAM Journal on Computing, vol.31, issue.3, pp.683-705, 2001.

E. M. Mccreight, A space-economical suffix tree construction algorithm, J. ACM, vol.23, issue.2, pp.262-272, 1976.

A. Meir and J. Moon, On an asymptotic method in enumeration, Journal of Combinatorial Theory (A), vol.51, pp.77-89, 1989.
DOI : 10.1016/0097-3165(89)90078-2
URL : https://doi.org/10.1016/0097-3165(89)90078-2

H. Mohamed and P. Robert, Dynamic tree algorithm, The Annals of Applied Probability, vol.20, issue.1, pp.26-51, 2010.

B. Morcrette, Master's thesis, Thèse de master MPRI Paris, 2010.

B. Morcrette, Combinatoire analytique et modèles d'urnes, Univ. Paris, vol.6, 2013.

R. Morris, Counting large numbers of events in small registers, Communications of the ACM, vol.21, issue.10, pp.840-842, 1978.

D. R. Morrison, Patricia-practical algorithm to retrieve information coded in alphanumeric, Journal of the ACM, vol.15, issue.4, pp.514-534, 1968.
DOI : 10.1145/321479.321481

R. Neininger, Probabilistic analysis of algorithms, stochastic fixed-point equations and ideal metrics, 2006.

R. Neininger and L. Rüschendorf, A general limit theorem for recursive algorithms and combinatorial structures, Annals of Applied Probability, vol.14, pp.378-418, 2004.

J. Neveu, Martingales à temps discret, 1972.

J. Neveu, Arbres et processus de Galton-Watson. Annales de l'Institut Henri Poincaré, vol.22, pp.199-207, 1986.

S. Nilsson and G. Karlsson, Ip-address lookup using lc-tries, IEEE journal on Selected Areas in Communications, vol.17, issue.6, pp.1083-1092, 1999.

N. E. Nörlund, Leçons sur les équations linéaires aux différences finies, Collection de monographies sur la théorie des fonctions. Gauthier-Villars, 1929.

N. E. Nörlund, Vorlesungen über Differenzenrechnung, 1954.

J. Norris, Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics, 1997.

A. Odlyzko, Periodic oscillations of coefficients of power series that satisfy functional equations, Advances in Mathematics, vol.44, issue.2, pp.180-205, 1982.

R. Otter, The number of trees, Annals of Mathematics, vol.40, issue.3, pp.583-599, 1948.

W. Panny, Deletions in random binary search trees : A story of errors, Journal of Statistical Planning and Inference, vol.140, issue.8, pp.2335-2345, 2010.

J. B. Paris, A. Vencovská, and G. M. Wilmers, A natural prior probability distribution derived from the propositional calculus, Annals of Pure and Applied Logic, vol.70, pp.243-285, 1994.

G. Park, H. Hwang, P. Nicodème, and W. Szpankowski, Profiles of tries, SIAM Journal on Computing, vol.38, issue.5, pp.1821-1880, 2009.

Y. Peres, Probability on trees : an introductory climb. École d'été de Saint-Flour, Lectures on Probability Theory and Statistics, 1997.

V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Lectures on Probability Theory and Statistics, 1995.

B. Pittel, On growing random binary trees, J. Math. Anal. Appl, vol.103, issue.2, pp.461-480, 1984.

B. Pittel, Paths in a Random Digital Tree : Limiting Distributions, Advances in Applied Probability, vol.18, pp.139-155, 1986.

B. Pittel, Note on the heights of random recursive trees and random m-ary search trees, Random Structures & Algorithms, vol.5, issue.2, pp.337-347, 1994.

G. Pólya, Sur quelques points de la théorie des probabilités, Annales de l'Institut Henri Poincaré, vol.1, pp.117-161, 1931.

G. Pólya and R. Read, Combinatorial enumeration of Groups, Graphs and Chemical Compounds, 1987.

T. Porter and I. Simon, Random insertion into a priority queue structure, IEEE Transactions on Software Engineering, SE, vol.1, issue.3, pp.292-298, 1975.

N. Pouyanne, Classification of large Pólya-Eggenberger urns with regard to their asymptotics, Discrete Mathematics and Theoretical Computer Science, AD, pp.275-286, 2005.

N. Pouyanne, An algebraic approach to Pólya processes, vol.44, pp.293-323, 2008.

N. Pouyanne, Urnes de Pólya. Mini-cours de Mahdia, 2012.

N. Pouyanne and . Transfert, col, inversion de Lagrange : ouvrir les boîtes noires, 2014.

H. Prodinger, How to select a loser, Discrete Mathematics, vol.20, pp.149-159, 1993.

B. Reed, The height of a random binary search tree, J. ACM, vol.50, issue.3, pp.306-332, 2003.

M. Régnier, A limiting distribution for quicksort. RAIRO, Theoretical Informatics and Applications, vol.23, pp.335-343, 1989.

E. Reingold, A note on 3-2 trees, Fibonacci Quarterly, pp.151-157, 1979.

D. Revuz and M. Yor, Continuous martingales and Brownian motion, 1999.

R. F. Rice, Some practical universal noiseless coding techniques, 1979.

M. Roberts, Almost sure asymptotics for the random binary search tree, DMTCS Proceedings, 21st International Meeting on Probabilistic, Combinatorial, and Asymptotic Methods in the Analysis of Algorithms (AofA'10), 2010.
URL : https://hal.archives-ouvertes.fr/hal-00459166

U. Rösler, A limit theorem for quicksort, Theoritical Informatics and Applications, vol.25, issue.1, pp.85-100, 1991.

U. Rösler and L. Rüschendorf, The contraction method for recursive algorithms, Algorithmica, vol.29, issue.1-2, pp.3-33, 2001.

M. Roux and B. Vallée, Information theory : Sources, Dirichlet series, and realistic analyses of data structures, Proceedings 8th International Conference Words, vol.63, pp.199-214, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01082040

D. Salomon, Data Compression : The Complete Reference, 2007.

P. Savický, Complexity and probability of some Boolean formulas, Combinatorics, Probability and Computing, vol.7, pp.451-463, 1998.

P. Savický and A. Woods, The number of Boolean functions computed by formulas of a given size, Random Structures & Algorithms, vol.13, pp.349-382, 1998.

G. Schaeffer, Bijective census and random generation of eulerian planar maps with prescribed degrees, Electronic Journal of Combinatorics, vol.4, issue.1, 1997.

R. Sedgewick, Algorithms in C : Fundamentals, Data Structures, Sorting, Searching, 1988.

R. Sedgewick and P. Flajolet, Introduction to the Analysis of Algorithms, 1996.

R. Sedgewick and K. Wayne, Algorithms, 2011.

R. Seidel and M. Sharir, Top-down analysis of path compression, SIAM Journal on Computing, vol.34, issue.3, pp.515-525, 2005.

B. W. Silverman, Density Estimation for Statistics and Data Analysis, 1986.

R. T. Smythe, Central limit theorems for urn models, Stochastic Processes and their Applications, vol.65, pp.115-137, 1996.

R. T. Smythe and H. Mahmoud, A survey of recursive trees, Theor. Probability and Math. Statist, vol.51, pp.1-27, 1995.

M. Sorensen and P. Urzyczyn, Lectures on the Curry-Howard isomorphism, 1998.

W. Szpankowski, Average Case Analysis of Algorithms on Sequences, 2001.

L. Takács, On the total height of random rooted binary trees, Journal of Combinatorial Theory (B), vol.61, pp.155-166, 1994.

R. E. Tarjan, Efficiency of a good but not linear set union algorithm, Journal of the ACM, vol.22, issue.2, pp.215-225, 1975.

B. Tsybakov and V. Mikhailov, Free synchronous packet access in a broadcast channel with feedback, Probl. Inform. Transmission, vol.14, pp.259-280, 1979.

K. Uchiyama, Spatial growth of a branching process of particles living in R d, Annals of Probability, vol.10, pp.896-918, 1982.

E. Ukkonen, On-line construction of suffix trees, Algorithmica, vol.14, issue.3, pp.249-260, 1995.

B. Vallée, Dynamical sources in information theory : Fundamental intervals and word prefixes, Algorithmica, vol.29, issue.1, pp.262-306, 2001.

B. Vallée, J. Clément, J. A. Fill, and P. Flajolet, The number of symbol comparisons in quicksort and quickselect, Part I, vol.5555, pp.750-763, 2009.

P. Weiner, Linear pattern matching algorithm, IEEE 14th Annual Symposium on Switching and Automata Theory, pp.1-11, 1973.

D. Welsh, Codes and Cryptography, 1990.

, Generating trees and the Catalan and Schröder numbers, Discrete Mathematics, vol.157, pp.363-374, 1995.

D. Williams, Probability with martingales. Cambridge Mathematical Textbooks, 1991.

J. W. Williams, Algorithm-232-heapsort, Communications of the ACM, vol.7, issue.6, pp.347-348, 1964.

A. Woods, Coloring rules for finite trees, and probabilities of monadic second order sentences, Random Structures & Algorithms, vol.10, pp.453-485, 1997.

W. Wu, Packet Forwarding Technologies, 2007.

A. Yao, On random 2-3 trees, Acta Informatica, vol.9, pp.159-170, 1978.