# Length minimizing graphs between a finite set of points

Consider a set of $$n$$ points in the plane. Among all the connected graphs (trees) $$T$$ in the plane that have these $$n$$ points among their vertices, I am looking to find one such that the sum of its edge lengths is minimum. Note that this question is different from Minimum Spanning Tree as we allow $$T$$ to have vertices other than the given $$n$$ points. For example, if $$n=3$$, there are 2 possibilities: Let $$A,B,C$$ be the three given points. If one of the angles in the triangle $$ABC$$, say $$ABC$$ is bigger than $$120^\circ$$ then AB+BC is the minimizing tree. If all the angles are less than $$120^\circ$$, let $$P$$ be the point within the triangle $$ABC$$ such that the angles $$APB$$, $$BPC$$, $$CPA$$ are all $$120^\circ$$. Then the tree $$T$$ with the set of vertices $$A,B,C,P$$ and edges $$AP,BP,CP$$ is the length minimizing one.

Is this problem worked out before?