An Asymptotic Analysis of Labeled and Unlabeled k-Trees

Abstract

In this paper we provide a systematic treatment of several shape parameters of (random) k-trees. Our research is motivated by many important algorithmic applications of k-trees in the context of tree-decomposition of a graph and graphs of bounded tree-width. On the other hand, k-trees are also a very interesting object from the combinatorial point of view. For both labeled and unlabeled k-trees, we prove that the number of leaves and more generally the number of nodes of given degree satisfy a central limit theorem with mean value and variance that are asymptotically linear in the size of the k-tree. In particular we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers \(U_k(n)\) of unlabeled k-trees of size n are asymptotically given by \(U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}\), where \(c_k> 0\) and \(\rho _{k}>0\) denotes the radius of convergence of the generating function \(U(z)=\sum _{n\ge 0} U_k(n) z^n\).

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Acknowledgments

We thank the anonymous reviewers for helpful suggestions on the first version of this paper.

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Correspondence to Emma Yu Jin.

Additional information

The first author is partially supported by the Austrian Science Fund FWF, Project F50-02. The second author is supported by the German Research Foundation DFG, Project JI 207/1-1 and Austrian Research Fund FWF, SFB F50 Algorithmic and Enumerative Combinatorics.

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Drmota, M., Jin, E.Y. An Asymptotic Analysis of Labeled and Unlabeled k-Trees. Algorithmica 75, 579–605 (2016). https://doi.org/10.1007/s00453-015-0039-1

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Keywords

  • k-trees
  • Generating function
  • Singularity analysis
  • Central limit theorem

Mathematics Subject Classification

  • 05A16
  • 05A15