The shape of unlabeled rooted random trees

We consider the number of nodes in the levels of unlabelled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabelled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process.

[1]  J. Moon,et al.  On the Altitude of Nodes in Random Trees , 1978, Canadian Journal of Mathematics.

[2]  Philippe Chassaing,et al.  The height and width of simple trees , 2000 .

[3]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[4]  R. W. Robinson,et al.  Twenty-step algorithm for determining the asymptotic number of trees of various speces , 1975, Journal of the Australian Mathematical Society.

[5]  L. Bruce Richmond,et al.  The Distribution of Heights of Binary Trees and Other Simple Trees , 1993, Combinatorics, Probability and Computing.

[6]  R. Otter.  The Number of Trees , 1948 .

[7]  J. Pitman.  The SDE solved by local times of a brownian excursion or bridge derived from the height profile of a random tree or forest , 1999 .

[8]  Lajos Takács,et al.  Conditional limit theorems for branching processes , 1991 .

[9]  Michael Drmota,et al.  The distribution of nodes of given degree in random trees , 1999, J. Graph Theory.

[10]  HwangHsien-Kuei.  Profiles of random trees: Plane-oriented recursive trees , 2007 .

[11]  Robert W. Robinson,et al.  The distribution of degrees in a large random tree , 1975, Discret. Math..

[12]  Michael Drmota,et al.  The Width of Galton-Watson Trees Conditioned by the Size , 2004, Discret. Math. Theor. Comput. Sci..

[13]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[14]  G. Hooghiemstra,et al.  On the explicit form of the density of Brownian excursion local time , 1982 .

[15]  de Ng Dick Bruijn,et al.  THE AVERAGE HEIGHT OF PLANTED PLANE TREES , 1972 .

[16]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[17]  Michael Drmota,et al.  On the profile of random trees , 1997, Random Struct. Algorithms.

[18]  D. Aldous.  Stochastic Analysis: The Continuum random tree II: an overview , 1991 .

[19]  Bernhard Gittenberger.  Convergence of branching processes to the local time of a Bessel process , 1998 .

[20]  B'en'edicte Haas,et al.  Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees , 2010, 1003.3632.

[21]  L. Takács.  On the distribution of the number of vertices in layers of random trees , 1991 .

[22]  M. Drmota,et al.  The Profile of Binary Search Trees , 2001 .

[23]  Andrew M. Odlyzko,et al.  Bandwidths and profiles of trees , 1985, J. Comb. Theory, Ser. B.

[24]  Pierre Nicod,et al.  Average profiles, from tries to suffix-trees , 2005 .

[25]  Michael Drmota,et al.  On Nodes Of Given Degree In Random Trees , 1996 .

[26]  Michael Drmota,et al.  On Robson's convergence and boundedness conjectures concerning the height of binary search trees , 2004, Theor. Comput. Sci..

[27]  Guy Louchard,et al.  Average Profile of the Generalized Digital Search Tree and the Generalized Lempel-Ziv Algorithm , 1999, SIAM J. Comput..

[28]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[29]  Hsien-Kuei Hwang,et al.  WIDTH AND MODE OF THE PROFILE FOR SOME RANDOM TREES OF LOGARITHMIC HEIGHT , 2006, math/0607119.

[30]  Guy Louchard,et al.  The Brownian excursion multi-dimensional local time density , 1999, Journal of Applied Probability.

[31]  G. Pólya.  Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen , 1937 .

[32]  Hsien-Kuei Hwang,et al.  Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees , 2006, Algorithmica.

[33]  Philippe Flajolet,et al.  The Average Height of Binary Trees and Other Simple Trees , 1982, J. Comput. Syst. Sci..

[34]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[35]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[36]  Bernhard Gittenberger.  On the Profile of Random Forests , 2002 .

[37]  P. Flajolet,et al.  The height of random binary unlabelled trees , 2008, 0807.2365.

[38]  Jean-François Marckert,et al.  The CRT is the scaling limit of unordered binary trees , 2009, Random Struct. Algorithms.

[39]  Ronald C. Read,et al.  Graph theory and computing , 1972 .

[40]  Bernhard Gittenberger,et al.  Nodes of large degree in random trees and forests , 2006, Random Struct. Algorithms.

[41]  Michael Drmota,et al.  Bimodality and Phase Transitions in the Profile Variance of Random Binary Search Trees , 2005, SIAM J. Discret. Math..

[42]  Hsien-Kuei Hwang,et al.  Profiles of random trees: correlation and width of random recursive trees and binary search trees , 2005, Advances in Applied Probability.

[43]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[44]  A. Meir,et al.  On Nodes of Given Out-Degree in Random Trees , 1992 .

[45]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions , 1920, Nature.

[46]  J. W. Cohen,et al.  Brownian Excursion, the M/M/1 Queue and Their Occupation Times , 1981, Mathematics of Operations Research.

[47]  M. Drmota.  Random Trees: An Interplay between Combinatorics and Probability , 2009 .

[48]  Piet Van Mieghem,et al.  On the covariance of the level sizes in random recursive trees , 2002, Random Struct. Algorithms.

[49]  Hsien-Kuei Hwang,et al.  Profiles of random trees: Plane‐oriented recursive trees , 2007, Random Struct. Algorithms.