Abstract
In this paper we consider the class of binary trees and present the results of a comprehensive experimental study on the four most representative algorithms for drawing trees, one for each of the following treedrawing approaches: Separation-Based, Path-based, Level-based, and Ringed Circular Layout. We establish a simpler, more intuitive format for storing binary trees in files and create a large suite of randomlygenerated, unbalanced, complete, AVL, Fibonacci, and molecular combinatory binary trees of various sizes. Our study is therefore conducted on randomly-generated, unbalanced, and AVL binary trees with between 100 and 50, 000 nodes, on Fibonacci trees Tn for n = 1, 2, ..., 45, 46 (143 to 46, 367 nodes), on complete binary trees of size 2−1 for n = 7, 8, ..., 15, 16 (127 to 65, 535 nodes), and on molecular combinatory binary trees with between 133 and 50, 005 nodes. Our study yields 70 charts comparing the performance of the drawing algorithms with respect to ten quality measures, namely Area, Aspect Ratio, Size, Total Edge Length, Average Edge Length, Maximum Edge Length, Uniform Edge Length, Angular Resolution, Closest Leaf, and Farthest Leaf. None of the algorithms has been found to be the best in all categories. This observation leads us to create an adaptive system that determines the type of a binary tree and then selects an algorithm to draw the tree depending upon the specified quality measures. Currently, our adaptive tree drawing system recognizes all six types of binary trees and all ten measures included in our experimental study. Under our settings, our adaptive tree drawing system outperforms any system using a single binary tree drawing algorithm. Submitted: April 2007 Reviewed: June 2007 Revised: August 2007 Accepted: October 2007 Final: November 2007 Published: June 2008 Article type: Regular Paper Communicated by: S. Kobourov E-mail addresses: rusu@rowan.edu (Adrian Rusu) santia09@students.rowan.edu (Confesor Santiago) 132 Rusu & Santiago Grid Drawings of Binary Trees
Highlights
Trees are ubiquitous data-structures, arising in a variety of applications such as Software Engineering, Business Administration, Knowledge Representation, and Web-site Design and Visualization.Visualizing a tree can enhance a user’s ability in understanding its structure
In this paper, we present an experimental study of some well-known algorithms for drawing binary trees
Angular Resolution: the minimum angle between any two edges in the drawing. It is widely accepted [1, 2, 23, 24] that small values of the area, size, total edge length, average edge length, maximum edge length, and uniform edge length are related to the perceived aesthetic appeal and visual effectiveness of the drawing
Summary
Trees are ubiquitous data-structures, arising in a variety of applications such as Software Engineering (class and module hierarchies), Business Administration (organization charts), Knowledge Representation (isa hierarchies), and Web-site Design and Visualization (structure of a Web-site). Within our experimental setting, we have performed a comparative study of four representative algorithms for planar straight-line grid drawings of binary trees, one for each of the following distinct approaches: separationbased algorithm by Garg and Rusu [16], path-based algorithm by Chan et al [8], level-based algorithm by Reingold and Tilford [25], and ringed circular layout algorithm by Teoh and Ma [33]. Our experimental setting consists of (i) a simpler, more intuitive format for storing binary trees in files; (ii) save/load routines for generating binary trees to files and for uploading binary trees from files, respectively; (iii) a large suite of randomly-generated, unbalanced, complete, AVL, Fibonacci, and molecular combinatory binary trees of various sizes; (iv) ten quality measures: area, aspect ratio, size, total edge length, average edge length, maximum edge length, uniform edge length, angular resolution, closest leaf, and farthest leaf
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