The reduction takes as input an instance of the Steiner tree problem: a weighted graph, with a subset of its vertices selected as terminals. The goal of the Steiner tree problem is to connect these terminals by a tree whose weight is as small as possible. To transform this problem into an instance of the -minimum spanning tree problem, attach to each terminal a tree of zero-weight edges with a large number of vertices per tree. (For a graph with vertices and terminals, they use added vertices per tree.) Then, they ask for the -minimum spanning tree in this augmented graph with . The only way to include this many vertices in a -spanning tree is to use at least one vertex from each added tree, for there are not enough vertices remaining if even one of the added trees is missed. However, for this choice of , it is possible for -spanning tree to include only as few edges of the original graph as are needed to connect all the terminals. Therefore, the -minimum spanning tree must be formed by combining the optimal Steiner tree with enough of the zero-weight edges of the added trees to make the total tree size large enough.