Asymptotic distribution of two-protected nodes in ternary search trees

Abstract

We study protected nodes in $m$ - ary search trees, by putting them in context of generalised Po lya 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 Po lya 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 $ .