Algorithms for drawing binary trees in the plane

doi:10.1016/S0020-0190(98)00049-0

Information Processing Letters

Volume 66, Issue 3, 15 May 1998, Pages 133-139

https://doi.org/10.1016/S0020-0190(98)00049-0 Get rights and content

Abstract

In this paper, we present several algorithms for drawing binary trees in the plane under various measures such as area, aspect ratio, the sum of edge lengths, and the number of bends per edge.

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Optimizing area and aspect ratio in straight-line orthogonal tree drawings
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Algorithms for drawing graphs: an annotated bibliography
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Cited by (7)

Optimizing area and aspect ratio in straight-line orthogonal tree drawings
2002, Computational Geometry: Theory and Applications
We investigate the problem of drawing an arbitrary n-node binary tree orthogonally and upwardly in an integer grid using straight-line edges. We show that one can simultaneously achieve good area bounds while also allowing the aspect ratio to be chosen as a fixed constant or a parameter under the user's control. In addition, we show that one can also achieve an additional desirable aesthetic criterion, which we call “subtree separation”. Our drawings require O(nlogn) area, which we show is optimal to within constant factors in the worst case (i.e. there are trees that need $Ω(n log n)$ area for any upward orthogonal straight-line drawing with good aspect ratio). An improvement for non-upward drawings is briefly mentioned.
Area-efficient algorithms for straight-line tree drawings
2000, Computational Geometry: Theory and Applications
We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment.
Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from $O (n log n)$ (Crescenzi et al., 1992) to $O (n loglog n)$ . This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type.
Our results are briefly summarized as follows. Let $T$ be a bounded-degree tree with $n$ vertices. Firstly, we show that $T$ admits an upward straight-line drawing with area $O (n loglog n)$ . If $T$ is binary, we can obtain an $O (n loglog n)$ -area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has $O (n/ log n)$ bends in total. Secondly, we present $O (n loglog n)$ -area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.
Tree Drawings Revisited
2020, Discrete and Computational Geometry
Improved Upper and Lower Bounds for LR Drawings of Binary Trees
2020, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Improved Upper and Lower Bounds for LR Drawings of Binary Trees
2019, arXiv
Tree drawings revisited
2018, Leibniz International Proceedings in Informatics, LIPIcs

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^☆: This is supported in part by KOSEF under the Grant No. 94-0100-04-01-3.

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Algorithms for drawing binary trees in the plane☆

Abstract

Optimizing area and aspect ratio in straight-line orthogonal tree drawings

Algorithms for drawing graphs: an annotated bibliography

Comput. Geom.