%0 Journal Article
%T Asymptotic distribution of two-protected nodes in ternary search trees
%A Cecilia Holmgren
%A Svante Janson
%J Mathematics
%D 2014
%I arXiv
%X We study protected nodes in $m$-ary search trees, by putting them in context of generalised P\'olya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m $-ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $ m $; we conjecture that the method yields an asymptotically normal distribution for all $m\leq 26$. The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler P\'olya urn (that is similar to the one that has earlier been used to study the total number of nodes in $ m $-ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $ m\leq 26 $.
%U http://arxiv.org/abs/1403.5573v1