The problem of designing a spanning tree on an underlying graph to minimize the flow costs of a given set of traffic demands is considered. Several new classes of valid inequalities are developed for the problem. Tests on 10-node problem instances show that the addition of these inequalities results in integer solutions for a significant majority of the instances without requiring any branching. In the remaining cases, root gaps of less than 1% from the optimal solutions are realized. For 30-node problem instances, the inequalities substantially reduce the number of nodes explored in the branch-and-bound tree, resulting in significantly reduced computational times. Optimal solutions are reported for problems with 30 nodes, 60 edges, fully dense traffic matrices, and Euclidean distance-based flow costs. Problems with such flow costs are well-known to be a very difficult class of problems to solve. Using the inequalities substantially improves the performance of a variable-fixing heuristic. Some polyhedral issues relating to the strength of these inequalities are also discussed.