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 $$U_k(n)$$Uk(n) of unlabeled k-trees of size n are asymptotically given by $$U_k(n) \sim c_k n^{-5/2}\rho _{k}^{-n}$$Uk(n)~ckn-5/2?k-n, where $$c_k> 0$$ck>0 and $$\rho _{k}>0$$?k>0 denotes the radius of convergence of the generating function $$U(z)=\sum _{n\ge 0} U_k(n) z^n$$U(z)=?n?0Uk(n)zn.