The max+sum spanning tree (MSST) problem is to determine a spanning tree T whose combined weight Formula omitted is minimum for a given edge-weighted undirected network G(V, E, c, w). This problem can be solved within Formula omitted time, where m and n are the numbers of edges and nodes, respectively. An inverse MSST problem (IMSST) aims to determine a new weight vector Formula omitted so that a given Formula omitted becomes an optimal MSST of the new network Formula omitted. The IMSST problem under weighted Formula omitted norm is to minimize the modification cost Formula omitted, where q(e) is a cost modifying the weight w(e). We first provide some optimality conditions of the problem. Then we propose a strongly polynomial time algorithm in Formula omitted time based on a binary search and a greedy method. There are O(m) iterations and we solve an MSST problem under a new weight in each iteration. Finally, we perform some numerical experiments to show the efficiency of the algorithm.