Aequationes mathematicae

, Volume 92, Issue 2, pp 311–353

Fringe analysis of plane trees related to cutting and pruning

• Benjamin Hackl
• Clemens Heuberger
• Sara Kropf
• Helmut Prodinger
Open Access
Article

Abstract

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.

Keywords

Plane trees Pruning Tree reductions Central limit theorem Narayana polynomials

Mathematics Subject Classification

05A16 05C05 05A15 05A19 60C05

Notes

Acknowledgements

Open access funding provided by University of Klagenfurt

References

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)
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)
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)
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)
8. 8.
Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)
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)
10. 10.
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
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)
15. 15.
Janson, S.: Random cutting and records in deterministic and random trees. Random Struct. Algorithms 29(2), 139–179 (2006)
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)
17. 17.
Kemp, R.: A note on the stack size of regularly distributed binary trees. BIT 20(2), 157–162 (1980)
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))
19. 19.
Meir, A., Moon, J.W.: Cutting down random trees. J. Aust. Math. Soc. 11, 313–324 (1970)
20. 20.
NIST Digital library of mathematical functions. http://dlmf.nist.gov/, 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)
22. 22.
Prodinger, H.: The height of planted plane trees revisited. Ars Combin. 16(B), 51–55 (1983)
23. 23.
The SageMath Developers: SageMath Mathematics Software (Version 7.4) (2016). http://www.sagemath.org
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))
25. 25.
Wagner, S.: Central limit theorems for additive tree parameters with small toll functions. Combin. Probab. Comput. 24, 329–353 (2015)
26. 26.
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1996). (Reprint of the fourth (1927) edition)
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)

Authors and Affiliations

2. 2.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan
3. 3.Department of Mathematical SciencesStellenbosch UniversityStellenboschSouth Africa