Every Set of Disjoint Line Segments Admits a Binary Tree

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist configurations of line segments where any such tree requires degree 3. We provide an O(n log n) time algorithm for constructing such a tree, and show that this is optimal.