Trees are usually drawn planar, i.e. without any crossings. In this paper we investigate the area requirement of (non-upward) planar straight-line drawings of binary trees. Let T be a binary tree with n vertices. We show that T admits a planar straight-line grid drawing with area O(n) and with any prespecified aspect ratio in the range [1; n ], where is a constant such that 0 < 1. We also show that such a drawing can be constructed in O(n log n) time.

We present several simple methods to construct planar, strictly upward, strongly orderpreserving, straight-line drawings of any n-node binary tree. In particular, it is shown that O(n 1+" ) area is always sucient for an arbitrary constant " > 0. Key Words. Graph drawing, Trees. 1

A track graph is a graph with its vertex set partitioned into horizontal levels. It is track planar if there are permutations of the vertices on each level such that all edges can be drawn as weak monotone curves without crossings. The novelty and generalisation over level planar graphs is that horizontal edges connecting consecutive vertices on the same level are allowed. We show that track planarity can be reduced to level planarity in linear time. Hence, there are time algorithms for the track planarity test and for the computation of a track planar embedding.

Abstract — A mathematical evaluation and comparison of the space-efficiency of various 2D graphical representations of tree structures is presented. As part of the evaluation, a novel metric called the mean area exponent is introduced that quantifies the distribution of area across nodes in a tree representation, and that can be applied to a broad range of different representations of trees. Several representations are analyzed and compared by calculating their mean area exponent as well as the area they allocate to nodes and labels. Our analysis inspires a set of design guidelines as well as a few novel tree representations that are also presented. Index Terms—Tree visualization, graph drawing, efficiency metrics. 1

Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution. 1

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

We show that there exist series-parallel graphs requiring Ω(n2 √ logn) area in any straightline or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. [Information Processing Letters, 2003]. Second, we show a family of series-parallel graphs requiring Ω(2 √ logn) width and Ω(2 √ logn) height in any straight-line or poly-line grid drawing. Combining the two results, the Ω(n2 √ logn) area lower bound is achieved. 2 1

An upward drawing of a rooted tree T is a planar straight-line drawing of T where the vertices of T are placed on a set of horizontal lines, called layers, such that for each vertex u of T, no child of u is placed on a layer vertically above the layer on which u has been placed. In this paper we give alinear-time algorithm toobtainan upwarddrawing ofagiven rooted tree T on the minimum number of layers. Moreover, if the given tree T is not rooted, we can select a vertex r of T in linear time such that an upward drawing of T rooted at r would require the minimum number of layers among all the upward drawings of T with any of its vertices as the root. We also extend our results on a rooted tree to give an algorithm for an upwarddrawing of arooted ordered tree. To the best of our knowledge, there is no previous algorithm for obtaining an upward drawing of a tree on the minimum number of layers.

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