Frank Ruskey

Generating Binary Trees Lexicographically

We represent a binary tree by the level numbers of its leaves from left to right. Thus every binary tree of n leaves corresponds to a sequence of n numbers. We first give the necessary and sufficient conditions for a sequence to represent a binary tree; then we give an algorithm for generating all the feasible sequences lexicographically as a list. Also, algorithms are developed to determine the position of a given sequence, or to generate the sequence of a given position. Finally, it is shown that the average time per sequence generated is constant (independent of the length of the sequence).

On Rotations and the Generation of Binary Trees

The rotation graph, Gn, has vertex set consisting of all binary trees with n nodes. Two vertices are connected by an edge if a single rotation will transform one tree into the other. We provide a simpler proof of a result of Lucas that Gn, contains a Hamilton path. Our proof deals directly with the pointer representation of the binary tree. This proof provides the basis of an algorithm for generating all binary trees that can be implemented to run on a pointer machine and to use only constant time between the output of successive trees. Ranking and unranking algorithms are developed for the ordering of binary trees implied by the generation algorithm. These algorithms have time complexity O(n2) (arithmetic operations). We also show strong relationships amongst various representations of binary trees and amongst binary tree generation algorithms that have recently appeared in the literature.

Drawing Area-Proportional Venn and Euler Diagrams

We consider the problem of drawing Venn diagrams for which each region’s area is proportional to some weight (e.g., population or percentage) assigned to that region. These area-proportional Venn diagrams have an enhanced ability over traditional Venn diagrams to visually convey information about data sets with interacting characteristics. We develop algorithms for drawing area-proportional Venn diagrams for any population distribution over two characteristics using circles and over three characteristics using rectangles and near-rectangular polygons; modifications of these algorithms are then presented for drawing the more general Euler diagrams. We present results concerning which population distributions can be drawn using specific shapes. A program to aid further investigation of area-proportional Venn diagrams is also described.

Three-dimensional graph drawing

AbstractGraph drawing research has been mostly oriented toward two-dimensional drawings. This paper describes an investigation of fundamental aspects of three-dimensional graph drawing. In particular we give three results concerning the space required for three-dimensional drawings.We show how to produce a grid drawing of an arbitraryn-vertex graph with all vertices located at integer grid points, in ann×2n×2n grid, such that no pair of edges cross. This grid size is optimal to within a constant. We also show how to convert an orthogonal two-dimensional drawing in anH×V integer grid to a three-dimensional drawing with

Three-Dimensional Graph Drawing

\left\lceil {\sqrt H } \right\rceil \times \left\lceil {\sqrt H } \right\rceil \times V

Generating t -ary trees in A-order

volume. Using this technique we show, for example, that three-dimensional drawings of binary trees can be computed with volume

Simple Combinatorial Gray Codes Constructed by Reversing Sublists

\left\lceil {\sqrt n } \right\rceil \times \left\lceil {\sqrt n } \right\rceil \times \left\lceil {\log n} \right\rceil