We present two algorithms for finding Steiner trees in graphs, where the cost of a tree equals to the number of edges. In the case when the set of given points forms a node cover of the graph, we achieve the approximation ratio of 1 + 1/7 - 1/160. In the case when the set of given points forms a dominating set, we achieve 4/3. This is joint work with Dr. Martin Furer.