Aequationes mathematicae

, Volume 92, Issue 2, pp 311–353 | Cite as

Fringe analysis of plane trees related to cutting and pruning

  • Benjamin HacklEmail author
  • Clemens Heuberger
  • Sara Kropf
  • Helmut Prodinger
Open Access


Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.


Plane trees Pruning Tree reductions Central limit theorem Narayana polynomials 

Mathematics Subject Classification

05A16 05C05 05A15 05A19 60C05 



Open access funding provided by University of Klagenfurt


  1. 1.
    Callan, D.: Kreweras’s Narayana number identity has a simple Dyck path interpretation (2012). arXiv:1203.3999 [math.CO]
  2. 2.
    Chen, W.Y.C., Deutsch, E., Elizalde, S.: Old and young leaves on plane trees. Eur. J. Combin. 27(3), 414–427 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    de Bruijn, N.G., Knuth, D.E., Rice, S.O.: The Average Height of Planted Plane Trees. Graph theory and computing, pp. 15–22. Academic Press, New York (1972)zbMATHGoogle Scholar
  4. 4.
    de Chaumont, M.V.: Nombre de Strahler des arbres, languages algébrique et dénombrement de structures secondaires en biologie moléculaire. Doctoral thesis, Université de Bordeaux I (1985)Google Scholar
  5. 5.
    Drmota, M.: Random Trees. Springer, Wien (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Drmota, M.: Trees, Handbook of Enumerative Combinatorics, Discrete Mathematics and Applications, pp. 281–334. CRC Press, Boca Raton (2015)Google Scholar
  7. 7.
    Flajolet, P., Odlyzko, A.: The average height of binary trees and other simple trees. J. Comput. Syst. Sci. 25(2), 171–213 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Flajolet, P., Raoult, J.-C., Vuillemin, J.: The number of registers required for evaluating arithmetic expressions. Theor. Comput. Sci. 9(1), 99–125 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hackl, B., Heuberger, C., Prodinger, H.: Reductions of binary trees and lattice paths induced by the register function (2016).
  12. 12.
    Hackl, B., Kropf, S., Prodinger, H.: Iterative cutting and pruning of planar trees. In: Proceedings of the Fourteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO) (Philadelphia PA), SIAM, pp. 66–72 (2017)Google Scholar
  13. 13.
    Heuberger, C., Kropf, S.: Higher dimensional quasi-power theorem and Berry–Esseen inequality (2016). arXiv:1609.09599 [math.PR]
  14. 14.
    Hwang, H.-K.: On convergence rates in the central limit theorems for combinatorial structures. Eur. J. Combin. 19, 329–343 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Janson, S.: Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29(2), 139–179 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Janson, S.: Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees. Random Struct. Algorithms 48(1), 57–101 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kemp, R.: A note on the stack size of regularly distributed binary trees. BIT 20(2), 157–162 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kirschenhofer, P., Prodinger, H.: Further results on digital search trees. Theor. Comput. Sci. 58(1–3), 143–154 (1988). (Thirteenth International Colloquium on Automata, Languages and Programming (Rennes, 1986))MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Meir, A., Moon, J.W.: Cutting down random trees. J. Aust. Math. Soc. 11, 313–324 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    NIST Digital library of mathematical functions., Release 1.0.13 of 2016-09-16, 2016, Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. eds
  21. 21.
    Panholzer, A.: Cutting down very simple trees. Quaest. Math. 29(2), 211–227 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Prodinger, H.: The height of planted plane trees revisited. Ars Combin. 16(B), 51–55 (1983)MathSciNetzbMATHGoogle Scholar
  23. 23.
    The SageMath Developers: SageMath Mathematics Software (Version 7.4) (2016).
  24. 24.
    Viennot, X.G.: A Strahler bijection between Dyck paths and planar trees. Discrete Math. 246(1–3), 317–329 (2002). (Formal Power Series and Algebraic Combinatorics (1999))MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wagner, S.: Central limit theorems for additive tree parameters with small toll functions. Combin. Probab. Comput. 24, 329–353 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1996). (Reprint of the fourth (1927) edition)CrossRefzbMATHGoogle Scholar
  27. 27.
    Zeilberger, D.: A bijection from ordered trees to binary trees that sends the pruning order to the Strahler number. Discrete Math. 82(1), 89–92 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für MathematikAlpen-Adria-Universität KlagenfurtKlagenfurtAustria
  2. 2.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan
  3. 3.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa

Personalised recommendations