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Asymptotics of Relaxed k-Ary Trees

Authors Manosij Ghosh Dastidar , Michael Wallner

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Author Details

Manosij Ghosh Dastidar
  • Institut für Diskrete Mathematik und Geometrie, TU Wien, Austria
Michael Wallner
  • Institut für Diskrete Mathematik und Geometrie, TU Wien, Austria

Funding

  • Ghosh Dastidar, Manosij: supported by the Austrian Science Fund (FWF): P 34142.
  • Wallner, Michael: supported by the Austrian Science Fund (FWF): P 34142.

Acknowledgements

We warmly thank the three referees for their useful feedback.

Cite As Get BibTex

Manosij Ghosh Dastidar and Michael Wallner. Asymptotics of Relaxed k-Ary 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. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.15

Abstract

A relaxed k-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree k. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed k-ary tree with n nodes for n → ∞. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term e^{c n^{1/3}} appears in all these cases. We also derive the recurrences for compacted k-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
Keywords
  • Asymptotic enumeration
  • stretched exponential
  • Airy function
  • directed acyclic graph
  • Dyck paths
  • compacted trees
  • minimal automata

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Supplementary Materials

References

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