Steiner minimal trees on the union of two orthogonal rectangles

Suppose R is a union of two subsets Rl and R2 whose Steiner minimal trees SMT(R 1) and SMT(R2) are known. The decomposition question is when the Steiner minimal tree SMT(R) for R is just the union of two Steiner minimal trees on R 1 and R2 In this paper a special case studied, that R 1 =bcda, R2=defg are two non-overlapping rectangles with a common vertex d so that a,d,e lie on one line. We conclude that SMT(R) has only two possible structures. We also two sufficient conditions for the decomposition SMT(R 1 U R 2) SMT(R 1) U SMT(R2) , and prove that under suitable assumptions of randomness, the probability of such a decomposition is 0.9679.