Weakly protected nodes in random binary search trees

Ezzat Mohammad Nezhad,Mehri Javanian,Ramin Imany Nabiyyi

doi:10.1051/ita/2022002

Abstract

Here, we derive the exact mean and variance of the number of weakly protected nodes (the nodes that are not leaves and at least one of their children is not a leaf) in binary search trees grown from random permutations. Furthermore, by using contraction method, we prove normal limit law for a properly normalized version of this tree parameter.

Highlights

Let Xn denote the number of weakly protected nodes in a random binary search tree of size n
We begin to prove the normality of limiting distribution of Xn
The proof was completed by applying the contraction method, which was first introduced by [9], in studying the Quicksort algorithm

Summary

Definitions

A random binary search tree is generated by P as follows. The keys are stored in the internal nodes of the tree. The root of the tree stores the first key p1. Note that a uniform probability distribution on permutations does not induce a uniform probability distribution on binary search trees [5]. A protected node is a node that is not a leaf and none of its children is a leaf. We study the number of weakly protected nodes in random binary search trees. The number of weakly protected nodes have only been studied for ordered trees in [10]

The expectation and variance

Limiting distribution