Drawing Trees:Radial Layout

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Radial Layout

A radial layout of a tree is a variation of a level drawing, where the levels are concentric circles c1, c2,..., ch(T ) around the root placed at the origin c0.

Figure 45.11 shows an example of a radial layout. The radius of a circle ci, i = 1, 2,... , h(T ), is given by an increasing function r(i). This type of layout is frequently used for representing a free tree.

A free tree is a tree without a specific root. To layout a free tree, a node is chosen as a fictitious root that minimizes the height of the resulting subtrees. A straightforward manner to obtain a radial layout of a tree is to modify the algorithms for level drawings as presented in Sections 45.3 and 45.4.

To guarantee that the resulting drawing is planar, the subtree T (v) of each node v is  drawn within an annulus wedge w(v). In order to permit edges with endpoints in w(v) to extend outside w(v) and thus to conflict with other edges, the nodes in the subtree T (v) are placed within a convex subset s(v) of w(v). Given the level l(v), the node v is placed on cl(v) . Suppose that the tangent to cl(v) through v meets the points a and b on cl(v)+1.

Then s(v) is chosen to be the unbounded region that is given by the line segment ab and the rays from the root of T and the node a and b. The children vj , j = 1, 2,... ,k of v are then arranged on cl(v)+1 within s(v) according to the number of leaves in T (vj ).

If the distance between consecutive circles ci, ci+1 i = 0, 1,... , h(T ) − 1, is equal, it can be easily shown that the area occupied by the layout is in

image

Different algorithms for the radial layout of trees have been presented by [1, 8, 9] depending on the choice of the root, the radii of the circles and the angle of the annulus wedge. Symmetry oriented algorithms have also been developed, see e.g. [16].

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