M. I?, Since Y is connected, there is a path ? n 1 +1 between y 2 and y without points of N . By compactness, there exists z ? ? n 1

C. Therefore-d, Consider finally S the set of equivalence classes on the set of Cauchy sequences of Y for the relation u ? v if and only if d C (u n , v n ) ? 0. Fix?YFix? Fix?Y the equivalence classes of the constant sequences of Y . We have thus proved that S \ ? Y can be indexed by a subset of N Y × [D], where D is the maximal degree of a point of X. By the classical construction of the completion

. Finally, We can define a sequence (? y n ) of Y ? N Y , such that for any n, ? y n = y n if y n ? Y and else?yelse? else?y n = x, where x is the element of N Y naturally associated to y n as in the proof of the second point. Y ? N Y is compact for the topology induced by d as a closed subset of a compact, Therefore, there exists a subsequence

R. Bibliographie1, J. Abraham, P. Delmas, and . Hoscheit, A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces, Electronic Journal of Probability, vol.18, issue.14, pp.1-21, 2013.

L. Addario-berry and N. Broutin, Total progeny in killed branching random walk. Probability theory and related fields, pp.265-295, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01220798

L. Addario-berry, N. Broutin, and C. Goldschmidt, Critical Random Graphs: Limiting Constructions and Distributional Properties, Electronic Journal of Probability, vol.15, issue.0, pp.741-775, 2010.
DOI : 10.1214/EJP.v15-772

URL : https://doi.org/10.1214/ejp.v15-772

L. Addario-berry, N. Broutin, and C. Goldschmidt, The continuum limit of critical random graphs. Probability Theory and Related Fields, pp.367-406, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00773370

L. Addario-berry, N. Broutin, C. Goldschmidt, and G. Miermont, The scaling limit of the minimum spanning tree of the complete graph, The Annals of Probability, vol.45, issue.5, 2013.
DOI : 10.1214/16-AOP1132

URL : https://hal.archives-ouvertes.fr/hal-00773360

L. Addario-berry and B. Reed, Minima in branching random walks. The Annals of Probability, pp.1044-1079, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00795281

E. Aïdékon, Convergence in law of the minimum of a branching random walk. The Annals of Probability, pp.1362-1426, 2013.

E. Aïdékon, J. Berestycki, É. Brunet, and Z. Shi, Branching Brownian motion seen from its tip. Probability Theory and Related Fields, pp.405-451, 2013.

E. Aïdékon, Tail asymptotics for the total progeny of the critical killed branching random walk, Electronic Communications in Probability, vol.15, issue.0, pp.522-533, 2010.
DOI : 10.1214/ECP.v15-1583

E. Aïdékon, Y. Hu, and O. Zindy, The precise tail behavior of the total progeny of a killed branching random walk. The Annals of Probability, pp.3786-3878, 2013.

D. Aldous, Power laws and killed branching random walk

D. Aldous, The continuum random tree I. The Annals of Probability, 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. The Annals of Probability, pp.248-289, 1993.

D. Aldous, Brownian excursions, critical random graphs and the multiplicative coalescent. The Annals of Probability, pp.812-854, 1997.

D. Aldous and J. Pitman, The standard additive coalescent, Annals of Probability, vol.26, pp.1703-1726, 1998.

G. Alsmeyer, J. D. Biggins, and M. Meiners, The functional equation of the smoothing transform. The Annals of Probability, pp.2069-2105, 2012.

A. Louis-pierre-arguin, N. Bovier, and . Kistler, The extremal process of branching brownian motion. Probability Theory and Related Fields, pp.535-574, 2013.

I. Armendariz, Dual fragmentation and multiplicative coagulation ; related excursion processes

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

J. Berestycki, É. Brunet, C. Simon, P. Harris, and . Mi?o?, Branching Brownian motion with absorption and the all-time minimum of branching Brownian motion with drift, Journal of Functional Analysis, vol.273, issue.6, pp.273-2017
DOI : 10.1016/j.jfa.2017.06.006

J. Bertoin, A fragmentation process connected to Brownian motion. Probability Theory and Related Fields, pp.289-301, 2000.

S. Bhamidi, A. Budhiraja, and X. Wang, The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs. Probability Theory and Related Fields, pp.3-4733, 2014.

D. John, A. E. Biggins, and . Kyprianou, Measure change in multitype branching, Advances in Applied Probability, vol.36, issue.2, pp.544-581, 2004.

B. Bollobás, Random graphs. Number 73, 2001.

N. Andrei, P. Borodin, and . Salminen, Handbook of Brownian Motion : Facts and Formulae, 2002.

M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves. Memoirs of the, p.44, 1983.

M. Bramson, J. Ding, and O. Zeitouni, Convergence in law of the maximum of nonlattice branching random walk. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, pp.1897-1924, 2016.

N. Broutin and J. Marckert, A new encoding of coalescent processes : applications to the additive and multiplicative cases. Probability Theory and Related Fields, pp.515-552, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01092562

S. , I. Burago, M. Burago, and Y. , A course in metric geometry, 2001.

B. Chauvin, Product martingales and stopping lines for branching brownian motion. The Annals of Probability, pp.1195-1205, 1991.
DOI : 10.1214/aop/1176990340

URL : http://doi.org/10.1214/aop/1176990340

B. Chauvin and M. Drmota, The Random Multisection Problem, Travelling Waves and the Distribution of the Height of m-Ary Search Trees, Algorithmica, vol.46, issue.3-4, pp.3-4299, 2006.
DOI : 10.1007/s00453-006-0107-7

B. Chauvin, T. Klein, J. Marckert, and A. Rouault, Martingales and Profile of Binary Search Trees, Electronic Journal of Probability, vol.10, issue.0, pp.420-435, 2005.
DOI : 10.1214/EJP.v10-257

URL : https://hal.archives-ouvertes.fr/hal-00138808

B. Chauvin and A. Rouault, Kpp equation and supercritical branching brownian motion in the subcritical speed area. application to spatial trees. Probability theory and related fields, pp.299-314, 1988.

B. Chauvin and A. Rouault, Connecting Yule process, bisection and binary search tree via martingales, Journal of Iranian Statistical Society, vol.3, issue.2, pp.88-116, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00138806

A. Earl, N. Coddington, and . Levinson, Theory of ordinary differential equations, 1955.

P. Corre, Number of particles absorbed in a bbm on the extinction event. arXiv preprint, 2016.

P. Corre, Oscillations in the height of the Yule tree and application to the binary search tree, Random Structures & Algorithms, vol.4, issue.1, pp.90-120, 2017.
DOI : 10.1145/765568.765571

J. Daryl and D. Daley, An introduction to the theory of point processes : volume II : general theory and structure, 2007.

L. Devroye, A note on the height of binary search trees, Journal of the ACM, vol.33, issue.3, pp.489-498, 1986.
DOI : 10.1145/5925.5930

L. Devroye, Branching processes in the analysis of the heights of trees, Acta Informatica, vol.24, issue.3, pp.277-298, 1987.
DOI : 10.1007/BF00265991

L. Devroye and B. Reed, On the Variance of the Height of Random Binary Search Trees, SIAM Journal on Computing, vol.24, issue.6, pp.1157-1162, 1995.
DOI : 10.1137/S0097539792237541

M. Drmota, An analytic approach to the height of binary search trees, Algorithmica, vol.11, issue.1-2, pp.89-119, 2001.
DOI : 10.1145/5925.5930

M. Drmota, An analytic approach to the height of binary search trees II, Journal of the ACM, vol.50, issue.3, pp.333-374, 2003.
DOI : 10.1145/765568.765572

M. Drmota, Profile and height of random binary search trees, Journal of Iranian Statistical Society, vol.3, issue.2, pp.117-138, 2004.

M. Drmota, Random trees : an interplay between combinatorics and probability, 2009.
DOI : 10.1007/978-3-211-75357-6

M. Richard and . Dudley, Real analysis and probability, 2002.

T. Duquesne, A limit theorem for the contour process of condidtioned galtonwatson trees. The Annals of Probability, pp.996-1027, 2003.

T. Duquesne and J. Gall, Probabilistic and fractal aspects of lévy trees. Probability Theory and Related Fields, pp.553-603, 2005.

P. Erd?s and A. Rényi, On random graphs, I, Publicationes Mathematicae (Debrecen), vol.6, pp.290-297, 1959.

P. Erd?s and A. Rényi, On the evolution of random graphs, Bull. Inst. Internat. Statist, vol.38, issue.4, pp.343-347, 1961.

W. Feller, An introduction to probability and its applications, 1971.

P. Flajolet and R. Sedgewick, Analytic combinatorics, 2009.
DOI : 10.1017/CBO9780511801655

URL : https://hal.archives-ouvertes.fr/inria-00072739

B. Haas and G. Miermont, Scaling limits of markov branching trees with applications to galton?watson and random unordered trees. The Annals of Probability, pp.2589-2666, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00464337

J. M. Hammersley, Postulates for subadditive processes. The Annals of Probability, pp.652-680, 1974.
DOI : 10.1214/aop/1176996611

URL : http://doi.org/10.1214/aop/1176996611

R. Hardy and S. C. Harris, A new formulation of the spine approach to branching diffusions. arXiv preprint math, 2006.

R. Hardy and S. C. Harris, A Spine Approach to Branching Diffusions with Applications to L p -Convergence of Martingales, Séminaire de Probabilités XLII, pp.281-330, 2009.
DOI : 10.1007/978-3-642-01763-6_11

J. W. Harris, S. C. Harris, and A. E. Kyprianou, Further probabilistic analysis of the Fisher???Kolmogorov???Petrovskii???Piscounov equation: one sided travelling-waves, Annales de l'Institut Henri Poincare (B) Probability and Statistics, pp.125-145, 2006.
DOI : 10.1016/j.anihpb.2005.02.005

C. Simon and . Harris, Travelling-waves for the FKPP equation via probabilistic arguments, Proceedings of the Royal Society of Edinburgh : Section A Mathematics, vol.129, issue.03, pp.503-517, 1999.

E. Hille, Analytic function theory, 1962.

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

L. Edward and . Ince, Ordinary differential equations, 1944.

J. Jabbour-hattab, Martingales and large deviations for binary search trees, Random Structures and Algorithms, vol.11, issue.2, pp.112-127, 2001.
DOI : 10.1016/0022-247X(84)90141-0

URL : http://www.math.uvsq.fr/~jabbour/submitRSA.ps

S. Janson, T. Luczak, and A. Rucinski, Random graphs
DOI : 10.1002/9781118032718

O. Kallenberg, Foundations of modern probability, 2006.
DOI : 10.1007/978-1-4757-4015-8

P. Douglas and . Kennedy, The distribution of the maximum brownian excursion, Journal of Applied Probability, vol.13, issue.2, pp.371-376, 1976.

H. Kesten, Branching brownian motion with absorption, Stochastic Processes and their Applications, pp.9-47, 1978.
DOI : 10.1016/0304-4149(78)90035-2

URL : https://doi.org/10.1016/0304-4149(78)90035-2

N. Andreï, I. G. Kolmogorov, N. S. Petrovskii, and . Piskunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Bjulleten' Moskovskogo Gosudarstvennogo Universiteta, vol.1, issue.7, pp.1-26, 1937.

A. E. Kyprianou, Travelling wave solutions to the KPP equation : alternatives to Simon Harris' probabilistic analysis, Annales de l'Institut Henri Poincaré (B) Probability and Statistics, pp.53-72, 2004.

P. Steven, T. A. Lalley, and . Sellke, A conditional limit theorem for the frontier of a branching brownian motion. The Annals of Probability, pp.1052-1061, 1987.

S. Lang, Complex analysis, 2013.

J. Gall, Random real trees, Annales de la facult?? des sciences de Toulouse Math??matiques, vol.15, issue.1, pp.35-62, 2006.
DOI : 10.5802/afst.1112

A. Mikhail and . Lifshits, Cyclic behavior of maxima in a hierarchical summation scheme. arXiv preprint, 2012.

Q. Liu, Fixed points of a generalized smoothing transformation and applications to the branching random walk, Advances in Applied Probability, vol.84, issue.01, pp.85-112, 1998.
DOI : 10.1017/S0001867800039902

Q. Liu, On generalized multiplicative cascades, Stochastic Processes and their Applications, pp.263-286, 2000.
DOI : 10.1016/S0304-4149(99)00097-6

URL : https://doi.org/10.1016/s0304-4149(99)00097-6

T. Madaule, Convergence in Law for the Branching Random Walk Seen from Its Tip, Journal of Theoretical Probability, vol.18, issue.3, pp.27-63, 2017.
DOI : 10.1214/ECP.v18-2390

M. Hosam and . Mahmoud, Evolution of random search trees, 1992.

P. Maillard, The number of absorbed individuals in branching Brownian motion with a barrier Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, pp.428-455, 2013.

P. Henry and . Mckean, Application of Brownian motion to the equation of Kolmogorov- Petrovskii-Piskunov, Communications on Pure and Applied Mathematics, vol.28, issue.3, pp.323-331, 1975.

G. Miermont, Tessellations of random maps of arbitrary genus, Annales scientifiques de l'??cole normale sup??rieure, vol.42, issue.5, pp.725-781, 2009.
DOI : 10.24033/asens.2108

URL : https://hal.archives-ouvertes.fr/hal-00200685

J. Neveu, Multiplicative Martingales for Spatial Branching Processes, Seminar on Stochastic Processes, pp.223-242, 1987.
DOI : 10.1007/978-1-4684-0550-7_10

J. Pitman, Enumerations of trees and forests related to branching processes and random walks. Microsurveys in discrete probability, pp.163-180, 1998.

J. Pitman, Coalescent Random Forests, Journal of Combinatorial Theory, Series A, vol.85, issue.2, pp.165-193, 1999.
DOI : 10.1006/jcta.1998.2919

URL : https://doi.org/10.1006/jcta.1998.2919

B. Pittel, On growing random binary trees, Journal of Mathematical Analysis and Applications, vol.103, issue.2, pp.461-480, 1984.
DOI : 10.1016/0022-247X(84)90141-0

URL : https://doi.org/10.1016/0022-247x(84)90141-0

G. Pólya and G. Szeg?, Problems and Theorems in Analysis, 1972.

B. Reed, The height of a random binary search tree, Journal of the ACM, vol.50, issue.3, pp.306-332, 2003.
DOI : 10.1145/765568.765571

I. Matthew and . Roberts, Almost sure asymptotics for the random binary search tree. arXiv preprint, 2010.

J. M. Robson, The height of binary search trees, Australian Computer Journal, vol.11, issue.4, pp.151-153, 1979.
URL : https://hal.archives-ouvertes.fr/hal-00307160

J. M. Robson, The asymptotic behaviour of the height of binary search trees, Australian Computer Science Communications, vol.4, issue.1, pp.88-98, 1982.

W. Rudin, Real and complex analysis, 1987.

T. Yang and Y. Ren, Limit theorem for derivative martingale at criticality w.r.t branching Brownian motion, Statistics & Probability Letters, vol.81, issue.2, pp.195-200, 2011.
DOI : 10.1016/j.spl.2010.11.007