We consider a new network design problem that generalizes the Hop and Diameter Constrained Minimum Spanning and Steiner Tree Problem as follows: given an edge-weighted undirected graph whose nodes are partitioned into a set of root nodes, a set of terminals and a set of potential Steiner nodes, find a minimum-weight subtree that spans all the roots and terminals so that the number of hops between each relevant node and an arbitrary root does not exceed a given hop limit H. The set of relevant nodes may be equal to the set of terminals, or to the union of terminals and root nodes. This paper presents theoretical and computational comparisons of flow-based vs. path-based mixed integer programming models for this problem. Disaggregation by roots is used to improve the quality of lower bounds of both models. To solve the problem to optimality, we implement branch-and-price algorithms for all proposed formulations. Our computational results show that the branch-and-price approaches based on path formulations outperform the flow formulations if the hop limit is not too loose.