A Tale of Two Trees: New Analysis for AVL Tree and Binary Heap

TL;DR

This paper offers new analytical insights into AVL trees and binary heaps, providing simpler proofs for their height and construction time, enhancing understanding of these fundamental data structures.

Contribution

It introduces new properties and potential function-based analysis methods that simplify and tighten bounds for AVL tree height and heap construction time.

Findings

AVL tree height is logarithmic, proven via new properties.

Bottom-up heap construction has a tight worst-case bound.

Simplified analysis techniques improve understanding of these data structures.

Abstract

In this paper, we provide new insights and analysis for the two elementary tree-based data structures - the AVL tree and binary heap. We presented two simple properties that gives a more direct way of relating the size of an AVL tree and the Fibonacci recurrence to establish the AVL tree's logarithmic height. We then give a potential function-based analysis of the bottom-up heap construction to get a simpler and tight bound for its worst-case running-time.