Asymptotic degree distribution in random recursive trees

The distributions of vertex degrees in random recursive trees and random plane recursive trees are shown to be asymptotically normal. Formulas are given for the asymptotic variances and covariances of the number of vertices with given outdegrees. We also give functional limit theorems for the evolution as vertices are added. The proofs use some old and new results about generalized Polya urn models. We consider generalized Polya urns with infinitely many types, but reduce them to the finite type case. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005

[1]  Philippe Flajolet,et al.  Varieties of Increasing Trees , 1992, CAAP.

[2]  K. Athreya,et al.  Embedding of Urn Schemes into Continuous Time Markov Branching Processes and Related Limit Theorems , 1968 .

[3]  Norman L. Johnson,et al.  Urn models and their application , 1977 .

[4]  Luc Devroye,et al.  Limit Laws for Local Counters in Random Binary Search Tree , 1991, Random Struct. Algorithms.

[5]  Nicolas Pouyanne,et al.  m-ary Search trees when m ge 27: A strong asymptotics for the space requirements , 2004, Random Struct. Algorithms.

[6]  Anatol Rapoport,et al.  Distribution of nodes of a tree by degree , 1970 .

[7]  C. C. Heyde,et al.  On the number of terminal vertices in certain random trees with an application to stemma construction in philology , 1982, Journal of Applied Probability.

[8]  Hosam M. Mahmoud,et al.  The Joint Distribution of Elastic Buckets in Multiway Search Trees , 1994, SIAM J. Comput..

[9]  Feifang Hu,et al.  Gaussian approximation theorems for urn models and their applications , 2002 .

[10]  Robert T. Smythe,et al.  Central limit theorems for urn models , 1996 .

[11]  Hosam M. Mahmoud.  The size of random bucket trees via urn models , 2004, Acta Informatica.

[12]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[13]  Hsien-Kuei Hwang,et al.  Phase changes in random m‐ary search trees and generalized quicksort , 2001, Random Struct. Algorithms.

[14]  Svante Janson,et al.  Functional limit theorems for multitype branching processes and generalized Pólya urns , 2004 .

[15]  Jerzy Szymanski,et al.  On the Structure of Random Plane-oriented Recursive Trees and Their Branches , 1993, Random Struct. Algorithms.

[16]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[17]  Feifang Hu,et al.  Asymptotic theorems for urn models with nonhomogeneous generating matrices , 1999 .

[18]  Amitava Bagchi,et al.  Asymptotic Normality in the Generalized Polya–Eggenberger Urn Model, with an Application to Computer Data Structures , 1985 .

[19]  Hosam M. Mahmoud,et al.  Asymptitic Hoint Normality of Outdegrees of Nodes in Random Recursive Trees , 1992, Random Struct. Algorithms.