Exploring Werner Krandick's Binary Tree Jump Statistics

TL;DR

This paper uses experimental mathematics and symbolic computation to derive explicit formulas for Krandick's binary tree jump statistics, providing moments, variance, and asymptotic behaviors, including proofs of normality and non-normality.

Contribution

It introduces a computational approach to obtain detailed statistical properties of Krandick's binary tree jump statistics, extending prior ad hoc methods.

Findings

Derived closed-form weight-enumerators for the statistics.

Computed the first eight moments of the statistics.

Proved the asymptotic normality of the number of jumps and non-normality of the sum of jump distances.

Abstract

Twenty years ago, Werner Krandick defined two statistics on binary trees. The first one determines the number of jumps, when traversing the tree in depth-first-search, from a vertex to one closer to the root, and the second keeps tracks of the sum of the jump-distances. He used clever but ad hoc human-generated arguments to find explicit expressions for their expectations. In this methodological note, we illustrate the power of experimental mathematics and symbolic computation to do much more. We derive closed-form expressions for the actual weight-enumerators according to these statistics (from which not only the expectations, but also the variances, and as many higher moments as desired, can be obtained). We also actually give the first eight moments, and conjecture that the first statistic (number of jumps) is asymptotically normal, and prove that the second one (sum of jump distances) is definitely not. In this revised version we are happy to announce that Stephen Melczer and Tia Ruza fully proved the asymptotic normality, and we provide a link to their writeup.