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: \(O(n^{1.55})\) for maximum degree 4 trees; \(O(n^{1.22})\) for maximum degree 3 (binary) trees; and \(O(n^{1.142})\) for perfect binary trees.