Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T19:43:51.118Z Has data issue: false hasContentIssue false
  • English
  • Français

The Distribution of Patterns in Random Trees

Published online by Cambridge University Press:  01 January 2008

FRÉDÉRIC CHYZAK ,
MICHAEL DRMOTA ,
THOMAS KLAUSNER  and
GERARD KOK
FRÉDÉRIC CHYZAK
Affiliation:
INRIA-Rocquencourt, F-78153 Le Chesnay cedex, France (e-mail: [email protected])
MICHAEL DRMOTA
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/113, A-1040 Wien, Austria (e-mail: [email protected], [email protected])
THOMAS KLAUSNER
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/113, A-1040 Wien, Austria (e-mail: [email protected], [email protected])
GERARD KOK
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, NL-2628 CD Delft, The Netherlands (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let denote the set of unrooted labelled trees of size n and let ℳ be a particular (finite, unlabelled) tree. Assuming that every tree of is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of ℳ is asymptotically normal with mean value and variance asymptotically equivalent to μn and σ2n, respectively, where the constants μ>0 and σ≥0 are computable.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 1 , January 2008 , pp. 21 - 59
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aho, A. V., Sethi, R. and Ullman, J. D. (1986) Compilers: Principles, Techniques, and Tools, Addison-Wesley.Google Scholar
[2]Bender, E. A. and Richmond, L. B. (1983) Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions. J. Combin. Theory Ser. A 34 255–265.CrossRefGoogle Scholar
[3]Drmota, M. (1994) Asymptotic distributions and a multivariate Darboux method in enumeration problems. J. Combin. Theory Ser. A 67 169–184.CrossRefGoogle Scholar
[4]Drmota, M. (1997) Systems of functional equations. Random Struct. Alg. 10 103–124.3.0.CO;2-Z>CrossRefGoogle Scholar
[5]Drmota, M. and Gittenberger, B. (1999) The distribution of nodes of given degree in random trees. J. Graph Theory 31 227–253.3.0.CO;2-6>CrossRefGoogle Scholar
[6]Dershowitz, N. and Zaks, S. (1989) Patterns in trees. Discrete Appl. Math. 25 241–255.CrossRefGoogle Scholar
[7]Flajolet, P., Gourdon, X. and Martínez, C. (1997) Patterns in random binary search trees. Random Struct. Alg. 11 223–244.3.0.CO;2-2>CrossRefGoogle Scholar
[8]Flajolet, P. and Sedgewick, R. (2006) Analytic Combinatorics. http://algo.inria.fr/flajolet/Publications/books.htmlGoogle Scholar
[9]Flajolet, P. and Steyaert, J.-M. (1980) On the analysis of tree-matching algorithms. In Automata, Languages and Programming: Proc. Seventh Internat. Colloq., Noordwijkerhout, 1980 (Bhattacharjee, G. P., ed.), Vol. 85 of Lecture Notes in Computer Science, Springer, Berlin, pp. 208–219.Google Scholar
[10]Flajolet, P. and Steyaert, J.-M. (1980) On the analysis of tree-matching algorithms. In Trees in Algebra and Programming: Proc. 5th Lille Colloq., Lille, 1980, University of Lille I, pp. 22–40.Google Scholar
[11]Kok, G. (2005) Pattern distribution in various types of random trees. In 2005 International Conference on Analysis of Algorithms, pp. 223–230.CrossRefGoogle Scholar
[12]Kok, G. J. P. (2005) The distribution of patterns in random trees. Thesis, Institut für Diskrete Mathematik und Geometrie, TU Wien, Austria.Google Scholar
[13]Lalley, S. Random walks on infinite free products and infinite algebraic systems of generating functions. http://www.stat.uchicago.edu/~lalley/Papers.Google Scholar
[14]Meir, A. and Moon, J. W. (1978) On the altitude of nodes in random trees. Canad. J. Math. 30 997–1015.CrossRefGoogle Scholar
[15]Otter, R. (1948) The number of trees. Ann. of Math. (2) 49 583–599.CrossRefGoogle Scholar
[16]Robinson, R. W. and Schwenk, A. J. (1975) The distribution of degrees in a large random tree. Discrete Math. 12 359–372.CrossRefGoogle Scholar
[17]Ruciński, A. (1988) When are small subgraphs of a random graph normally distributed? Probab. Theory Rel. Fields 78 1–10.CrossRefGoogle Scholar
[18]Steyaert, J.-M. and Flajolet, P. (1983) Patterns and pattern-matching in trees: An analysis. Inform. Control 58 19–58.CrossRefGoogle Scholar