Tree Drawings Revisited

TL;DR

This paper advances the understanding of tree drawing area requirements by providing improved bounds for various types of tree drawings, including straight-line, upward, orthogonal, and order-preserving drawings, for trees of different degrees.

Contribution

The paper presents new upper bounds on the drawing area for multiple classes of tree drawings, improving upon longstanding bounds in the literature.

Findings

Improved area bound for straight-line drawings of trees: n2^{O(√(log log n log log log n))}

Enhanced area bounds for binary tree drawings: n2^{O(log* n)}

New bounds for orthogonal and order-preserving drawings of binary trees

Abstract

We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that 1. every tree of size $n$ (with arbitrarily large degree) has a straight-line drawing with area $n2^{O(\sqrt{\log\log n\log\log\log n})}$, improving the longstanding $O(n\log n)$ bound; 2. every tree of size $n$ (with arbitrarily large degree) has a straight-line upward drawing with area $n\sqrt{\log n}(\log\log n)^{O(1)}$, improving the longstanding $O(n\log n)$ bound; 3. every binary tree of size $n$ has a straight-line orthogonal drawing with area $n2^{O(\log^*n)}$, improving the previous $O(n\log\log n)$ bound by Shin, Kim, and Chwa (1996) and Chan, Goodrich, Kosaraju, and Tamassia (1996); 4. every binary tree of size $n$ has a straight-line order-preserving drawing with area $n2^{O(\log^*n)}$, improving the previous $O(n\log\log n)$ bound by Garg and Rusu (2003); 5. every binary tree of size $n$ has a straight-line orthogonal order-preserving drawing with area $n2^{O(\sqrt{\log n})}$, improving the $O(n^{3/2})$ previous bound by Frati (2007).