Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

TL;DR

This paper analyzes the probabilistic properties of canonical trees, compact prefix-free codes, and sums of unit fractions, revealing their distributional behaviors and concentration properties through a comprehensive probabilistic approach.

Contribution

It provides a probabilistic analysis of various parameters of canonical trees and codes, including their height, width, and leaf distribution, establishing normality and concentration results.

Findings

Height distribution is normal.

Number of summands follows a normal distribution.

Width exhibits concentration around its expectation.

Abstract

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).