Tree Drawings Revisited

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)

Quad general tree drawing algorithm and general trees characterization: towards an environment for the experimental study on general tree drawing algorithms

Information visualization produces (interactive) visual representations of abstract data to reinforce human cognition and perception; thus enabling the viewer to gain knowledge about the internal structure of the data and causal relationships in it. The visualization of information hierarchies is concerned with the presentation of abstract hierarchical information about relationships between various entities. It has many applications in diverse domains such as software engineering, information systems, biology, and chemistry. Information hierarchies are typically modeled by an abstract tree, where vertices are entities and edges represent relationships between entities. The aim of visualizing tree drawings is to automatically produce drawings of trees which clearly reflect the relationships of the information hierarchy. This thesis is primarily concerned with introducing the new general tree drawing algorithm Quad that produces good visually distinguishable angles, and a characterization of general trees which allows us to classify general trees into several types based on their characteristics. Both of these topics are part of building an experimental study environment for the evaluation of drawing algorithms for general trees. The main achievements of this thesis include: 1. A study on characterization of general trees that aims to classify them into several types. 2. A tree drawing algorithm that produces visually distinguishable angles for high degree general trees with user specified angular coefficient

How to fit a tree in a box

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with n nodes can be drawn on a vn by vn grid. We also show that testing whether a given rooted binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.Peer ReviewedPostprint (author's final draft

Upward Octagonal Drawings of Ternary Trees

We explore ways to embed a ternary tree in an integer coordinate grid such that the width of the drawing is minimized. We provide upper and lower bounds on the width requirement of planar, straight-line, upward, order-preserving drawings of ternary trees in an octagonal grid. We present a linear-time algorithm for constructing such octagonal grid drawings of any $n$-node ternary tree with $O(n^{0.68})$ width. (This bound can be improved to $O(n^{0.631})$ width in the so-called HVA-model.) For ideal octagonal grid drawings of complete $n$-node ternary trees, we provide an $\Omega(n^{0.411})$ width lower bound