Discrepancies of subtrees

TL;DR

This paper investigates the multicolour discrepancy of all subtrees in a tree, establishing bounds that relate discrepancy to the number of leaves, and introduces new notions of oriented and high-dimensional discrepancies with bounds.

Contribution

It provides the first bounds on the discrepancy of subtrees in trees, relating discrepancy to leaves, and introduces new discrepancy notions with bounds for these cases.

Findings

Discrepancy is linearly related to the number of leaves.

Bounds on discrepancy are asymptotically sharp.

Introduces and bounds oriented and high-dimensional discrepancies.

Abstract

We study multicolour, oriented and high-dimensional discrepancies of the set of all subtrees of a tree. As our main result, we show that the $r$-colour discrepancy of the subtrees of any tree is a linear function of the number of leaves $\ell$ of that tree. More concretely, we show that it is bounded by $\lceil(r-1)\ell/r\rceil$ from below and $\lceil(r-1)\ell/2\rceil$ from above, and that these bounds are asymptotically sharp. Motivated by this result, we introduce natural notions of oriented and high-dimensional discrepancies and prove bounds for the corresponding discrepancies of the set of all subtrees of a given tree as functions of its number of leaves.