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 Pólya 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 $m\le 26$. The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler Pólya 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 $m\le 26$.

References in corpus (1)

• Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees

Cited by in corpus (1)

• Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees