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 $U(z)=\sum _{n\ge 0} U_k(n) z^n$.