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 $ .
Highlights
Random trees are an important field within random graphs, partly due to the close relation between random trees and search algorithms in computer science
The mary search tree studied in this thesis, which is a generalisation of the binary search tree, has close connection to search algorithms
A node is called k-protected if it is has at least distance k to any leaf in the tree. These protected nodes are of interest given their relation to local properties close to the leaves of the random trees. They are related to so called fringe trees, which are small subtrees in the fringe of random trees
Summary
Random trees are an important field within random graphs, partly due to the close relation between random trees and search algorithms in computer science. The mary search tree studied in this thesis, which is a generalisation of the binary search tree, has close connection to search algorithms. A node is called k-protected if it is has at least distance k to any leaf in the tree These protected nodes are of interest given their relation to local properties close to the leaves of the random trees. In a paper by Cecilia Holmgren and Svante Janson, [12], a generalised Polya urn model was used which yielded asymptotic distributions on the number of protected nodes for the binary and ternary search tree.
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