Skip to main content
SpringerLink
Log in
Book cover

International Computer Science Symposium in Russia

CSR 2020: Computer Science – Theory and Applications pp 155–168Cite as

Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

  • Conference paper
  • First Online:

  • 634 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12159))

Abstract

There exists a wealth of literature concerning families of increasing trees, particularly suitable for representing the evolution of either data structures in computer science, or probabilistic urns in mathematics, but are also adapted to model evolutionary trees in biology. The classical notion of increasing trees corresponds to labeled trees such that, along paths from the root to any leaf, node labels are strictly increasing; in addition nodes have distinct labels. In this paper we introduce new families of increasingly labeled trees relaxing the constraint of unicity of each label. Such models are especially useful to characterize processes evolving in discrete time whose nodes evolve simultaneously. In particular, we obtain growth processes for biology much more adequate than the previous increasing models. The families of monotonic trees we introduce are much more delicate to deal with, since they are not decomposable in the sense of Analytic Combinatorics. New tools are required to study the quantitative statistics of such families. In this paper, we first present a way to combinatorially specify such families through evolution processes, then, we study the tree enumerations.

This work was also supported by the anr projects Metaconc ANR-15-CE40-0014.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    We take \(\phi _0 = 0\) in anticipation of our model.

References

  1. Bergeron, F., Flajolet, P., Salvy, B.: Varieties of increasing trees. In: Raoult, J.-C. (ed.) CAAP 1992. LNCS, vol. 581, pp. 24–48. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55251-0_2

    Google Scholar 

  2. Blieberger, J.: Monotonically labelled Motzkin trees. Discrete Appl. Math. 18(1), 9–24 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Bodini, O., Gardy, D., Gittenberger, B., Jacquot, A.: Enumeration of generalized BCI lambda-terms. Electr. J. Comb. 20(4), P30 (2013)

    MathSciNet  MATH  Google Scholar 

  4. Bodini, O., Genitrini, A., Gittenberger, B., Wagner, S.: On the number of increasing trees with label repetitions. Discrete Math. (2019, in press). https://doi.org/10.1016/j.disc.2019.111722

  5. Bodini, O., Genitrini, A., Naima, M.: Ranked Schröder trees. In: 2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 13–26. SIAM (2019)

    Google Scholar 

  6. Flajolet, P., Gourdon, X., Martínez, C.: Patterns in random binary search trees. Random Struct. Algorithms 11(3), 223–244 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  8. Genitrini, A., Gittenberger, B., Kauers, M., Wallner, M.: Asymptotic enumeration of compacted binary trees of bounded right height. J. Comb. Theory Ser. A 172 (2020). https://doi.org/10.1016/j.jcta.2019.105177

  9. Henrici, P.: Applied and Computational Complex Analysis, Volume 2. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley, Hoboken (1974)

    MATH  Google Scholar 

  10. Kirschenhofer, P.: On the average shape of monotonically labelled tree structures. Discrete Appl. Math. 7(2), 161–181 (1984)

    MathSciNet  MATH  Google Scholar 

  11. Mendez, M.A.: Combinatorial differential operators in: Faa di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees. Technical report. arXiv:1610.03602 [math.CO], Cornell University (2016)

  12. Morris, K.: On parameters in monotonically labelled trees. In: Drmota, M., Flajolet, P., Gardy, D., Gittenberger, B. (eds.) Mathematics and Computer Science III, pp. 261–263. Birkhäuser Basel (2004)

    Google Scholar 

  13. Morris, K.: Contributions to the analysis of increasing trees and other families of trees. Ph.D. thesis, University of the Witwatersrand, Johannesburg, South Africa (2005)

    Google Scholar 

  14. Odlyzko, A.M.: Periodic oscillations of coefficients of power series that satisfy functional equations. Adv. Math. 44(2), 180–205 (1982)

    MathSciNet  MATH  Google Scholar 

  15. Prodinger, H., Urbanek, F.J.: On monotone functions of tree structures. Discrete Appl. Math. 5(2), 223–239 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Pure and Applied Mathematics, Vol. XIV, Interscience Publishers Wiley, New York (1965)

    Google Scholar 

Download references

Acknowledgment

We thank Stephan Wagner for a fruitful discussion about the relationship of an involved proof of this paper and the article [8]. Furthermore we are grateful for the anonymous reviewers whose comments and suggestions helped improving and clarifying this manuscript.

Author information

Authors and Affiliations

  1. Université Sorbonne Paris Nord, Laboratoire d’Informatique de Paris Nord, CNRS, UMR 7030, 93430, Villetaneuse, France

    Olivier Bodini, Mehdi Naima & Alexandros Singh

  2. Sorbonne Université, CNRS, Laboratoire d’Informatique de Paris 6 -LIP6- UMR 7606, 75005, Paris, France

    Antoine Genitrini

Authors
  1. Olivier Bodini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antoine Genitrini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mehdi Naima
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alexandros Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Antoine Genitrini or Mehdi Naima .

Editor information

Editors and Affiliations

  1. University of Trier, Trier, Germany

    Henning Fernau

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Bodini, O., Genitrini, A., Naima, M., Singh, A. (2020). Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-50026-9_11

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-50025-2

  • Online ISBN: 978-3-030-50026-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation