Improved Upper and Lower Bounds for LR Drawings of Binary Trees

TL;DR

This paper improves the bounds on the width of LR drawings of binary trees, narrowing the gap between the best known upper and lower bounds for such planar straight-line upward drawings.

Contribution

It presents tighter upper and lower bounds for the width of LR drawings of binary trees, advancing the understanding of their geometric complexity.

Findings

Upper bound improved to O(n^{0.437})

Lower bound improved to Ω(n^{0.429})

Narrowed the gap between bounds for LR drawings

Abstract

In SODA'99, Chan introduced a simple type of planar straight-line upward order-preserving drawings of binary trees, known as LR drawings: such a drawing is obtained by picking a root-to-leaf path, drawing the path as a straight line, and recursively drawing the subtrees along the paths. Chan proved that any binary tree with $n$ nodes admits an LR drawing with $O(n^{0.48})$ width. In SODA'17, Frati, Patrignani, and Roselli proved that there exist families of $n$-node binary trees for which any LR drawing has $\Omega(n^{0.418})$ width. In this note, we improve Chan's upper bound to $O(n^{0.437})$ and Frati et al.'s lower bound to $\Omega(n^{0.429})$.