Improved bounds for drawing trees on fixed points with L-Shaped edges

Abstract

Let T be an n-node tree of maximum degree 4, and let P be a set of n points in the plane with no two points on the same horizontal or vertical line. It is an open question whether T always has a planar drawing on P such that each edge is drawn as an orthogonal path with one bend (an “L-shaped” edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set P for which such drawings are possible to: O(n1.55) for maximum degree 4 trees; O(n1.22) for maximum degree 3 (binary) trees; and O(n1.142) for perfect binary trees. Drawing ordered trees with L-shaped edges is harder—we give an example that cannot be done and a bound of O(n log n) points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.