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The height and width of simple trees

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Part of the book series: Trends in Mathematics ((TM))

Abstract

The limit law of the couple height-width for simple trees can be seen as a consequence of deep results of Aldous, Drmota and Gittenberger, and Jeulin. We give here an elementary proof in the case of binary trees

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References

  1. D. Aldous, (1991) The continuum random tree II: An overview, Stochastic analysis, Proc. Symp., Durham. UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 23–70.

    Article  MathSciNet  Google Scholar 

  2. D. Aldous, (1993) The continuum random tree III,Ann. of Probab. 21, No.1, 248–289.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Aldous, (1998) Brownian excursion conditionned on its local time, Elect. Comm. in Probab., 3, 79–90.

    MathSciNet  MATH  Google Scholar 

  4. P. Biane, M. Yor, (1987) Valeurs principales associées aux temps locaux browniens, Bull. Sci. Maths 111, 23–101.

    MathSciNet  MATH  Google Scholar 

  5. P. Biane, J. Pitman, M. Yor, (1999) Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, (seehttp://www.stat.berkeley.edu/~pitman//~pitman/)

    Google Scholar 

  6. P. Billingsley, (1968) Convergence of Probability Measures. John Wiley & Sons, Inc., New York-London-Sydney.

    MATH  Google Scholar 

  7. P. Chassaing, J.F. Marckert, (1999) Parking functions, empirical processes and the width of rooted labeled trees, preprint Elie Cartan, Université Nancy I.

    Google Scholar 

  8. P. Chassaing, J.F. Marckert, M. Yor, (1999) A Stochastically Quasi-Optimal Search Algorithm for the Maximum of the Simple Random Walk, to appear in Ann. of App. Prob.

    Google Scholar 

  9. K.L. Chung, (1976) Excursions in Brownian motion,Ark. För Math., 14, 155–177.

    Article  MATH  Google Scholar 

  10. T. H. Cormen, C. E. Leiserson, & R. L. Rivest, (1990) Introduction to algorithms, MIT Press, Cambridge, MA; McGraw-Hill Book Co., New York.

    MATH  Google Scholar 

  11. C. Donati-Martin, (1999) Some remarks about the identity in law for the Bessel bridge \( \int_{0}^{1} {\frac{{ds}}{{r\left( s \right)}}} \mathop{{{\text{ }} = }}\limits^{{\left( {law} \right)}} 2su{{p}_{{s1}}}r\left( s \right)\), prepub. Toulouse.

    Google Scholar 

  12. M. Drmota, B. Gittenberger, (1997) On the profile of random trees, Random Structures Algorithms 10, no. 4, 421–451.

    Article  MathSciNet  Google Scholar 

  13. M. Drmota, B. Gittenberger, (2000) The width of Galton-Watson trees,preprint.

    Google Scholar 

  14. R.T. Durrett, D.L. Iglehart, (1977) Functionals of Brownian meander and Brownian excursion, Ann. Probab. 5, 130–135.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Flajolet, A. Odlyzko, (1982) The average height of binary trees and other simple trees, J. Comp. and Sys. Sci., Vol. 25, No.2.

    Google Scholar 

  16. I.I. Gikhman, A.V.Skorohod, (1969) Introduction to the theory of random processes, W.B. Saunders, Philadelphia.

    Google Scholar 

  17. W. Gutjahr, G. Ch. Pflug, (1992) The asymptotic contour process of a binary tree is Brownian excursion, Stochastic Proc. Appl. 41, 69–90.

    Article  MathSciNet  MATH  Google Scholar 

  18. Th. Jeulin, (1980) Semi-martingales et grossissement d’une filtration,Lecture Notes in Mathematics, 833, Springer-Verlag.

    Google Scholar 

  19. D.G. Kendall, (1951) Some problems in the theory of queues, J. of the Roy. Stat. Soc. B 13, 151–185.

    MathSciNet  Google Scholar 

  20. G. Louchard, (1984) Kac’s formula, Levy’s local time and Brownian excursion, J.Appl. Prob. 21, 479–499.

    Article  MathSciNet  MATH  Google Scholar 

  21. A.M. Odlyzko, (1995) Search for the maximum of a random walk, Ran. Struct. Alg., Vol. 6, p. 275–295.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Pitman, M. Yor, (1996) Decomposition at the maximum for excursions and bridges of one-dimensional diffusions, Ito’s stochastic calculus and probability theory. Springer. 293–310.

    Chapter  Google Scholar 

  23. A. Renyi, G. Szekeres, (1967) On the height of trees,J. Aust. Math. Soc. 7, 497–507.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. P. Stanley, (1999) Enumerative combinatorics. Vol. 2,Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge.

    Book  Google Scholar 

  25. L. Takács, (1993) Limit distributions for queues and random rooted trees, J. Appl. Math. Stoch. Ana., 6, No.3, p.189–216.

    Article  Google Scholar 

  26. D. Williams, (1990) Brownian motion and the Riemann zeta-function, Disorder in physical systems, Clarendon press, Oxford, 361–372.

    Google Scholar 

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© 2000 Springer Basel AG

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Chassaing, P., Marckert, J.F., Yor, M. (2000). The height and width of simple trees. In: Gardy, D., Mokkadem, A. (eds) Mathematics and Computer Science. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8405-1_2

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  • DOI: https://doi.org/10.1007/978-3-0348-8405-1_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9553-8

  • Online ISBN: 978-3-0348-8405-1

  • eBook Packages: Springer Book Archive

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