A vertex v of a connected graph G = (V, E) is called a branch vertex if its degree is greater than two. Pertaining to branch vertices, this paper studies two optimization problems having roots in the domain of optical networks. The first one, referred to as MBV, seeks a spanning tree T of G with the minimum number of branch vertices, whereas the second problem, referred to as MDS, seeks a spanning tree T of G with the minimum degree sum of branch vertices. Both MBV and MDS are NP-Hard. Two heuristics approaches are presented for each problem. The first approach is a problem specific heuristic, whereas the latter one is a hybrid ant-colony optimization algorithm. Computational results show the effectiveness of our proposed approaches.