An Asymptotic Analysis of Unlabeled k-Trees
نویسندگان
چکیده
In this paper we solve the asymptotic counting problem for unlabeled k-trees. By applying a proper singularity analysis of generating functions we show that the numbers Un of unlabeled k-trees of size n are asymptotically given by Un ∼ ckn(ρ1k) , where ck > 0 and ρ1k > 0 denotes the radius of convergence of the generating function U(z) = ∑ n≥0 Unz . Furthermore 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 number of hedra where a hedron is a (k + 1)-clique in a k-tree.
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