Document Open Access Logo

On the Number of Distinct Fringe Subtrees in Binary Search Trees

Author Stephan Wagner

PDF

File

Thumbnail PDF
PDF
LIPIcs.AofA.2024.13.pdf
  • Filesize: 0.63 MB
  • 11 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Data compression
Keywords
  • Fringe subtrees
  • binary search trees
  • tree compression
  • minimal DAG
  • asymptotics

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to (c₁n)/(log n) for a constant c₁ ≈ 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.

Cite As Get BibTex

Stephan Wagner. On the Number of Distinct Fringe Subtrees in Binary Search Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.13

Author Details

Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Austria
  • Department of Mathematics, Uppsala University, Sweden

Funding

Supported by the Swedish research council (VR), grant 2022-04030.

References

  1. Alfred V. Aho, Ravi Sethi, and Jeffrey D. Ullman. Compilers: Principles, Techniques, and Tools. Addison-Wesley series in computer science / World student series edition. Addison-Wesley, 1986. Google Scholar
  2. Olivier Bodini, Antoine Genitrini, Bernhard Gittenberger, Isabella Larcher, and Mehdi Naima. Compaction for two models of logarithmic-depth trees: analysis and experiments. Random Structures Algorithms, 61(1):31-61, 2022. URL: https://doi.org/10.1002/rsa.21056.
  3. Mireille Bousquet-Mélou, Markus Lohrey, Sebastian Maneth, and Eric Noeth. XML compression via DAGs. Theory of Computing Systems, 57(4):1322-1371, 2015. URL: https://doi.org/10.1145/2448496.2448506.
  4. Randal E. Bryant. Symbolic boolean manipulation with ordered binary-decision diagrams. ACM Computing Surveys, 24(3):293-318, 1992. URL: https://doi.org/10.1145/136035.136043.
  5. Peter Buneman, Martin Grohe, and Christoph Koch. Path queries on compressed XML. In Johann Christoph Freytag et al., editors, Proceedings of the 29th Conference on Very Large Data Bases, VLDB 2003, pages 141-152. Morgan Kaufmann, 2003. URL: https://doi.org/10.1016/B978-012722442-8/50021-5.
  6. Luc Devroye. On the richness of the collection of subtrees in random binary search trees. Information Processing Letters, 65(4):195-199, 1998. URL: https://doi.org/10.1016/S0020-0190(97)00206-8.
  7. Michael Drmota. Random Trees: An Interplay Between Combinatorics and Probability. Springer, 1st edition, 2009. Google Scholar
  8. James Allen Fill. On the distribution of binary search trees under the random permutation model. Random Structures & Algorithms, 8(1):1-25, 1996. URL: https://doi.org/10.1002/(SICI)1098-2418(199601)8:1<1::AID-RSA1>3.0.CO;2-1.
  9. Philippe Flajolet, Xavier Gourdon, and Conrado Martínez. Patterns in random binary search trees. Random Structures & Algorithms, 11(3):223-244, 1997. URL: https://doi.org/10.1002/(SICI)1098-2418(199710)11:3<223::AID-RSA2>3.0.CO;2-2.
  10. Philippe Flajolet, Paolo Sipala, and Jean-Marc Steyaert. Analytic variations on the common subexpression problem. In Proceedings of the 17th International Colloquium on Automata, Languages and Programming, ICALP 1990, volume 443 of Lecture Notes in Computer Science, pages 220-234. Springer, 1990. URL: https://doi.org/10.1007/BFb0032034.
  11. Markus Frick, Martin Grohe, and Christoph Koch. Query evaluation on compressed trees. In Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science, LICS 2003, pages 188-197. IEEE Computer Society Press, 2003. URL: https://doi.org/10.1109/LICS.2003.1210058.
  12. Michael Fuchs. Limit theorems for subtree size profiles of increasing trees. Combinatorics, Probability and Computing, 21(3):412-441, 2012. URL: https://doi.org/10.1017/S096354831100071X.
  13. Zbigniew Gołebiewski, Abram Magner, and Wojciech Szpankowski. Entropy and optimal compression of some general plane trees. ACM Trans. Algorithms, 15(1):Art. 3, 23, 2019. URL: https://doi.org/10.1145/3275444.
  14. Cecilia Holmgren and Svante Janson. Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electronic Journal of Probability, 20:1-51, 2015. URL: https://doi.org/10.1214/EJP.v20-3627.
  15. Cecilia Holmgren, Svante Janson, and Matas Šileikis. Multivariate normal limit laws for the numbers of fringe subtrees in m-ary search trees and preferential attachment trees. Electron. J. Combin., 24(2):Paper No. 2.51, 49 pp., 2017. URL: https://doi.org/10.37236/6374.
  16. Svante Janson. Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton-Watson trees. Random Struct. Algorithms, 48(1):57-101, 2016. URL: https://doi.org/10.1002/rsa.20568.
  17. John C. Kieffer, En-Hui Yang, and Wojciech Szpankowski. Structural complexity of random binary trees. In Proceedings of the 2009 IEEE International Symposium on Information Theory, ISIT 2009, pages 635-639. IEEE, 2009. URL: https://doi.org/10.1109/ISIT.2009.5205704.
  18. Donald E. Knuth. The art of computer programming. Vol. 3. Addison-Wesley, Reading, MA, 1998. Google Scholar
  19. Dimbinaina Ralaivaosaona and Stephan Wagner. A central limit theorem for additive functionals of increasing trees. Combin. Probab. Comput., 28(4):618-637, 2019. URL: https://doi.org/10.1017/s0963548318000585.
  20. Dimbinaina Ralaivaosaona and Stephan G. Wagner. Repeated fringe subtrees in random rooted trees. In Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2015, pages 78-88. SIAM, 2015. URL: https://doi.org/10.1137/1.9781611973761.7.
  21. Louisa Seelbach Benkner and Markus Lohrey. Average case analysis of leaf-centric binary tree sources. In 43rd International Symposium on Mathematical Foundations of Computer Science, MFCS 2018, August 27-31, 2018, Liverpool, UK, pages 16:1-16:15, 2018. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.16.
  22. Louisa Seelbach Benkner and Stephan Wagner. Distinct fringe subtrees in random trees. Algorithmica, 84(12):3686-3728, 2022. URL: https://doi.org/10.1007/s00453-022-01013-y.
  23. Louisa Seelbach Benkner and Stephan G. Wagner. On the collection of fringe subtrees in random binary trees. In Yoshiharu Kohayakawa and Flávio Keidi Miyazawa, editors, LATIN 2020: Theoretical Informatics - 14th Latin American Symposium, São Paulo, Brazil, January 5-8, 2021, Proceedings, volume 12118 of Lecture Notes in Computer Science, pages 546-558. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-61792-9_43.
  24. Stephan Wagner. Central limit theorems for additive tree parameters with small toll functions. Combin. Probab. Comput., 24(1):329-353, 2015. URL: https://doi.org/10.1017/S0963548314000443.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact