Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-26T21:24:32.945Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit Theorems for Subtree Size Profiles of Increasing Trees

Published online by Cambridge University Press:  25 January 2012

MICHAEL FUCHS
MICHAEL FUCHS*
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan (e-mail: mfuchs@math.nctu.edu.tw)
Get access Rights & Permissions [Opens in a new window]

Abstract

Simple families of increasing trees were introduced by Bergeron, Flajolet and Salvy. They include random binary search trees, random recursive trees and random plane-oriented recursive trees (PORTs) as important special cases. In this paper, we investigate the number of subtrees of size k on the fringe of some classes of increasing trees, namely generalized PORTs and d-ary increasing trees. We use a complex-analytic method to derive precise expansions of mean value and variance as well as a central limit theorem for fixed k. Moreover, we propose an elementary approach to derive limit laws when k is growing with n. Our results have consequences for the occurrence of pattern sizes on the fringe of increasing trees.

Keywords

Primary 05C8060F0568W40
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 3 , May 2012 , pp. 412 - 441
Copyright
Copyright © Cambridge University Press 2012

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]Barabási, A.-L. and Albert, R. (1999) Emergence of scaling in random networks. Science 286 509–512.Google Scholar
[2]Bergeron, F., Flajolet, P. and Salvy, B. (1992) Varieties of increasing trees. In CAAP'92, Vol. 581 of Lecture Notes in Computer Science, Springer, pp. 24–48.CrossRefGoogle Scholar
[3]Chang, H. and Fuchs, M. (2010) Limit theorems for patterns in phylogenetic trees. J. Math. Biol. 60 481–512.CrossRefGoogle ScholarPubMed
[4]Chauvin, B., Drmota, M. and Jabbour-Hattab, J. (2001) The profile of binary search trees. Ann. Appl. Probab. 11 1042–1062.Google Scholar
[5]Chauvin, B., Klein, T., Marckert, J.-F. and Rouault, A. (2005) Martingales and profile of binary search trees. Electron. J. Probab. 10 420–435.CrossRefGoogle Scholar
[6]Chern, H.-H., Fuchs, M. and Hwang, H.-K. (2007) Phase changes in random point quadtrees. ACM Trans. Algorithms 3.Google Scholar
[7]Dennert, F. and Grübel, R. (2010) On the subtree size profile of binary search trees. Combin. Probab. Comput. 19 561–578.Google Scholar
[8]Devroye, L. (1998) On the richness of the collection of subtrees in random binary search trees. Inform. Process. Lett. 65 195–199.Google Scholar
[9]Drmota, M. and Gittenberger, B. (1997) On the profile of random trees. Random Struct. Alg. 10 421–451.Google Scholar
[10]Drmota, M. and Hwang, H.-K. (2005) Bimodality and phase transitions in the profile variance of random binary search trees. SIAM J. Discrete Math. 19 19–45.CrossRefGoogle Scholar
[11]Drmota, M. and Hwang, H.-K. (2005) Profile of random trees: correlation and width of random recursive trees and binary search trees. Adv. Appl. Probab. 37 321–341.Google Scholar
[12]Drmota, M., Janson, S. and Neininger, R. (2008) A functional limit theorem for the profile of search trees. Ann. Appl. Probab. 18 288–333.Google Scholar
[13]Drmota, M. and Spankowski, W. (2011) The expected profile of digital search trees. J. Combin. Theory Ser. A, 118, 1939–1965.CrossRefGoogle Scholar
[14]Elliott, P. D. T. A. (1979) Probabilistic Number Theory I: Central Limit Theorems, Springer.Google Scholar
[15]Feng, Q., Mahmoud, H. and Panholzer, A. (2008) Phase changes in subtree varieties in random recursive trees and binary search trees. SIAM J. Discrete Math. 22 160–184.Google Scholar
[16]Feng, Q., Mahmoud, H. and Su, C. (2007) On the variety of subtrees in a random recursive tree. Technical report, The George Washington University, Washington, DC.Google Scholar
[17]Feng, Q., Miao, B. and Su, C. (2006) On the subtrees of binary search trees. Chinese J. Appl. Probab. Statist. 22 304–310.Google Scholar
[18]Fill, J. and Kapur, N. (2004) Limiting distributions for additive functionals on Catalan trees. Theoretical Comput. Sci. 326 69–102.Google Scholar
[19]Flajolet, P., Gourdon, X. and Martinez, C. (1997) Patterns in random binary search trees. Random Struct. Alg. 11 223–244.Google Scholar
[20]Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.Google Scholar
[21]Fuchs, M. (2008) Subtree sizes in recursive trees and binary search trees: Berry–Esseen bounds and Poisson approximations. Combin. Probab. Comput. 17 661–680.Google Scholar
[22]Fuchs, M., Hwang, H.-K. and Neininger, R. (2006) Profiles of random trees: Limit theorems for random recursive trees and binary search trees. Algorithmica 46 367–407.Google Scholar
[23]Hwang, H.-K. (2007) Profiles of random trees: Plane-oriented recursive trees. Random Struct. Alg. 30 380–413.Google Scholar
[24]Kuba, M. and Panholzer, A. (2007) On the degree distribution of the nodes in increasing trees. J. Combin. Theory Ser. A 114 597–618.Google Scholar
[25]Panholzer, A. and Prodinger, H. (2007) The level of nodes in increasing trees revisited. Random Struct. Alg. 31 203–226.CrossRefGoogle Scholar
[26]Park, G., Hwang, H.-K., Nicodeme, P. and Szpankowski, W. (2009) Profiles of tries. SIAM J. Comput. 38 1821–1880.Google Scholar
[27]Rosenberg, N. A. (2006) The mean and variance of the number of r-pronged nodes and r-caterpillars in Yule-generated genealogical trees. Ann. Combin. 10 129–146.Google Scholar
[28]Smythe, R. T. and Mahmoud, H. (1995) A survey of recursive trees. Theory Probab. Math. Statist. 51 1–27.Google Scholar