Document Open Access Logo

Path Length and External Path Length in Random Trees

Authors Jacob Lundblad , Stephan Wagner

PDF

File

Thumbnail PDF
PDF
LIPIcs.AofA.2026.2.pdf
  • Filesize: 0.73 MB
  • 20 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Path length
  • external path length
  • Bienaymé-Galton-Watson trees
  • increasing trees

Metrics

Abstract

We consider two closely related concepts in rooted trees: the path length is the sum of all distances from the root to the vertices of the tree, while the external path length is the sum of all distances from the root to the leaves of the tree. Upon dividing by the number of vertices and leaves respectively, we obtain the average distance to the root. For two important classes of random trees, we show that the average distance of the root to a random leaf is almost the same as the average distance to a random vertex. For Bienaymé-Galton-Watson trees, the difference is bounded in probability. For three varieties of increasing trees (recursive trees, d-ary increasing trees, generalised plane-oriented recursive trees), on the other hand, the difference converges to a constant in probability.

Cite As Get BibTex

Jacob Lundblad and Stephan Wagner. Path Length and External Path Length in Random Trees. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.2

Author Details

Jacob Lundblad
  • Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Steyrergasse 30, 8010 Graz, Austria
  • Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden

Funding

  • Lundblad, Jacob: Supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.
  • Wagner, Stephan: Supported by the Swedish research council (VR), grant 2022-04030.

Supplementary Materials

References

  1. David Aldous. The continuum random tree. II. An overview. In Stochastic analysis (Durham, 1990), volume 167 of London Math. Soc. Lecture Note Ser., pages 23-70. Cambridge Univ. Press, Cambridge, 1991. URL: https://doi.org/10.1017/CBO9780511662980.003.
  2. François Bergeron, Philippe Flajolet, and Bruno Salvy. Varieties of increasing trees. In CAAP '92 (Rennes, 1992), volume 581 of Lecture Notes in Comput. Sci., pages 24-48. Springer, Berlin, 1992. URL: https://doi.org/10.1007/3-540-55251-0_2.
  3. Nicolas Broutin and Cecilia Holmgren. The total path length of split trees. Ann. Appl. Probab., 22(5):1745-1777, 2012. URL: https://doi.org/10.1214/11-AAP812.
  4. Robert P. Dobrow and James Allen Fill. Total path length for random recursive trees. Combin. Probab. Comput., 8(4):317-333, 1999. URL: https://doi.org/10.1017/S0963548399003855.
  5. Robert P. Dobrow and Robert T. Smythe. Poisson approximations for functionals of random trees. Random Structures Algorithms, 9(1-2):79-92, 1996. URL: https://doi.org/10.1002/(sici)1098-2418(199608/09)9:1/2<79::aid-rsa5>3.0.co;2-8.
  6. Michael Drmota. Random trees. SpringerWienNewYork, Vienna, 2009. An interplay between combinatorics and probability. URL: https://doi.org/10.1007/978-3-211-75357-6.
  7. Philippe Flajolet and Andrew Odlyzko. Singularity analysis of generating functions. SIAM J. Discrete Math., 3(2):216-240, 1990. URL: https://doi.org/10.1137/0403019.
  8. Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. URL: https://doi.org/10.1017/CBO9780511801655.
  9. B. V. Gnedenko and A. N. Kolmogorov. Limit distributions for sums of independent random variables. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. Google Scholar
  10. Svante Janson. Random cutting and records in deterministic and random trees. Random Structures Algorithms, 29(2):139-179, 2006. URL: https://doi.org/10.1002/rsa.20086.
  11. Svante Janson. Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Probab. Surv., 9:103-252, 2012. URL: https://doi.org/10.1214/11-PS188.
  12. M. Javanian and M. G. Vahidi-Asl. External path length of random m-oriented recursive. J. Natur. Sci. Math., 50(1-2):11-18, 2010. Google Scholar
  13. Mehri Javanian. On the external path length of random recursive k-ary trees. Ital. J. Pure Appl. Math., 31:21-26, 2013. Google Scholar
  14. Hosam M. Mahmoud. Limiting distributions for path lengths in recursive trees. Probab. Engrg. Inform. Sci., 5(1):53-59, 1991. URL: https://doi.org/10.1017/S0269964800001881.
  15. Hosam M. Mahmoud. Distances in random plane-oriented recursive trees. J. Comput. Appl. Math., 41(1-2):237-245, 1992. URL: https://doi.org/10.1016/0377-0427(92)90252-S.
  16. Jean-François Marckert and Abdelkader Mokkadem. The depth first processes of Galton-Watson trees converge to the same Brownian excursion. Ann. Probab., 31(3):1655-1678, 2003. URL: https://doi.org/10.1214/aop/1055425793.
  17. A. Meir and J. W. Moon. On the altitude of nodes in random trees. Canadian J. Math., 30(5):997-1015, 1978. URL: https://doi.org/10.4153/CJM-1978-085-0.
  18. Stephen Melczer. An invitation to analytic combinatorics - from one to several variables. Springer, Cham, 2021. URL: https://doi.org/10.1007/978-3-030-67080-1.
  19. J. W. Moon. The distance between nodes in recursive trees. In Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973), volume No. 13 of London Math. Soc. Lecture Note Ser., pages 125-132. Cambridge Univ. Press, London-New York, 1974. Google Scholar
  20. Götz Olaf Munsonius and Ludger Rüschendorf. Limit theorems for depths and distances in weighted random b-ary recursive trees. J. Appl. Probab., 48(4):1060-1080, 2011. URL: https://doi.org/10.1017/s0021900200008627.
  21. Alois Panholzer and Helmut Prodinger. Level of nodes in increasing trees revisited. Random Structures & Algorithms, 31(2):203-226, 2007. URL: https://doi.org/10.1002/rsa.20161.
  22. Mireille Régnier. A limiting distribution for quicksort. RAIRO Inform. Théor. Appl., 23(3):335-343, 1989. URL: https://doi.org/10.1051/ita/1989230303351.
  23. Uwe Rösler. A limit theorem for "Quicksort". RAIRO Inform. Théor. Appl., 25(1):85-100, 1991. URL: https://doi.org/10.1051/ita/1991250100851.
  24. Henning Sulzbach. On martingale tail sums for the path length in random trees. Random Structures Algorithms, 50(3):493-508, 2017. URL: https://doi.org/10.1002/rsa.20674.
  25. Jerzy Szymański. On a nonuniform random recursive tree. In Random graphs '85 (Poznań, 1985), volume 144 of North-Holland Math. Stud., pages 297-306. North-Holland, Amsterdam, 1987. Google Scholar
  26. Jerzy Szymański. On the complexity of algorithms on recursive trees. Theoret. Comput. Sci., 74(3):355-361, 1990. URL: https://doi.org/10.1016/0304-3975(90)90084-U.
  27. Lajos Takács. On the total heights of random rooted trees. J. Appl. Probab., 29(3):543-556, 1992. URL: https://doi.org/10.1017/s0021900200043370.
  28. Lajos Takács. The asymptotic distribution of the total heights of random rooted trees. Acta Sci. Math. (Szeged), 57(1-4):613-625, 1993. Google Scholar
  29. M. A. Tapia and B. R. Myers. Generation of concave node-weighted trees. IEEE Trans. Circuit Theory, CT-14:229-230, 1967. Google Scholar
  30. The Sage Developers. SageMath, the Sage Mathematics Software System, 2022. DOI 10.5281/zenodo.6259615. URL: https://www.sagemath.org.
  31. Yazhe Zhang. On the number of leaves in a random recursive tree. Braz. J. Probab. Stat., 29(4):897-908, 2015. URL: https://doi.org/10.1214/14-BJPS252.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact