The tree-drawing problem is to produce a `tidy' mapping of elements of a tree to points in the plane. In this paper, we derive an efficient algorithm for producing tidy drawings of trees. The specification, the starting point for the derivations, consists of a collection of intuitively appealing criteria satisfied by tidy drawings. The derivation shows constructively that these criteria completely determine the drawing. Indeed, the criteria completely determine a simple but inefficient algorithm for drawing a tree, which can be transformed into an efficient algorithm using just standard techniques and a small number of inventive steps. The algorithm consists of an upwards accumulation followed by a downwards accumulation on the tree, and is further evidence of the utility of these two higher-order tree operations.

We consider straight-line drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges fromeachnodetoitschildrenfromonetofive, andtofivepatterns: straight, Y, ψ, X, and full. The ψ–drawings generalize hv- or strictly upward drawings to ternary trees. Weshowthatcompleteternarytreeshavea ψ–drawingonasquareofsize O(n 1.262) and general ternary trees can be drawn within O(n 1.631) area. Bothboundsareoptimal.Sub–quadraticboundsarealsoobtainedfor X– pattern drawings of complete tetra trees, and for full–pattern drawings of complete penta trees, which are 4–ary and 5–ary trees. These results parallel and complement the ones of Frati [8] for straight–line orthogonal drawings of ternary trees. Moreover, we provide an algorithm for compacted straight–line drawings of penta trees on the hexagonal grid, such that the direction of the edges from a node to its children is given by our patterns and these edges have the same length. However, drawing trees on a hexagonal grid within a prescribed area or with unit length edges is NP–hard.

Abstract. We present almost linear area bounds for drawing complete trees on the octagonal grid. For 7-ary trees we establish an upper and lower bound of Θ(n 1.129) and for ternary trees the bounds of O(n 1.048) and Θ(n), where the latter needs edge bends. We explore the unit edge length and area complexity of drawing unordered trees on k-grids with k ∈ {4, 6, 8} and generalize the N P-hardness results of the orthogonal and hexagonal grid to the octagonal grid. 1

Swan is a data structure visualization system. It allows users to visualize the data structures and execution process of a C/C++ program. Swan views a data structure as a graph or collection of graphs. By “graph ” we mean general

In a box-drawing of a tree each node is drawn by a rectangular box of prescribed size, no two boxes overlap each other, all boxes corresponding to siblings of the tree have the same x-coordinate at their left sides, and a parent node is drawn at a given distance apart from its rst child. A box drawing of a tree is compact if it attains the minimum possible rectangular area enclosing the drawing. We give a linear-time algorithm for nding a compact box-drawing of a tree. A known algorithm does not always nd a compact box-drawing and takes time O(n ) if a tree has n nodes.

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We present almost linear area bounds for drawing trees on the octagonal grid. For complete 7-ary trees we establish an upper and lower bound of Θ(n 1.129) and for complete ternary trees the bounds of O(n 1.048) and Θ(n), where the latter needs edge bends. For arbitrary ternary trees we obtain an upper bound of O(n log log n) with bends and good aspect ratio by applying the recursive winding technique. We explore the unit edge length and area complexity of drawing unordered trees on k-grids with k ∈ {4, 6, 8} and generalize the N P-hardness results of the orthogonal grid to the octagonal and hexagonal grids. Submitted: