Steiner Minimal Trees

A Steiner minimal tree for given points A₁ , ⋯ ,Aₙ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum length the Steiner minimal tree may contain other vertices (Steiner points) beside A₁ , ⋯ ,Aₙ. We find conditions which simplify the task of constructing a Steiner minimal tree. Some of these use relationships with the easily constructed (ordinary) minimal tree which achieves minimum length among all trees having only A₁ , ⋯ ,Aₙ as vertices. Other questions concern the relative lengths of these two trees in extreme or typical cases. A review of the existing literature is included.