The paper is concerned with the problem of constructing a minimal cost weighted tree connecting a set of n given terminal vertices on a Euclidean plane. The authors prove that the problem is convex and that its solution is in the convex hull of the given terminal vertices and that the necessary and sufficient optimality conditions can be expressed by means of the simple optimality conditions of m two-dimensional Weber problems (m is the number of unknown extra vertices). The authors discuss a subgradient type algorithm in which the above optimality conditions are used as an ending rule. A better utilization of these conditions may consist in solving iteratively m Weber problems. Results obtained by this method show a fast convergence.