A Complicating Factor

The rule about there being just three angles doesn't apply if more than one link ends on one of the points. In the exceptional case below the Steiner tree is a combination of two three point trees. One has to make assumptions about the nature of the tree for a particular set of points. The assumptions affect the values of the objective function and there is no guarantee that the type of tree chosen will yield the global minimum. The reason for the exception in this case is that the point near the center lies on one of the links of the tree of the other four points. The central point would have to be far enough away from the central link for there to form a 120° angle in order for there to be another intermediate point. The result is that there may be some play in the structure of the trees.