Growing a Tree from Its Branches

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and the tree has no crossing edges. We also show that the maximum degree of a node in the minimum weight spanning tree formed by these line segments is seven, and that there exists a set of line segments achieving this bound. The time complexity of finding the minimum weight spanning tree is O(n log n). Finally, we show an (n logn) lower bound for this problem, establishing the optimality of the algorithm.