The Stackelberg Minimum Spanning Tree Game (StackMST) is defined in terms of a graph G = (V, B ∪ R), with two disjoint sets of edges, blue B and red R, and costs {c e ≥ 0 : e ∈ R} defined for the red edges. Once the leader of the game defines prices {p e : e ∈ B} to the blue edges, the follower chooses a minimum weight spanning tree (V, E T), at cost e∈B∩E T p e + e∈R∩E T c e. The goal is to find prices to maximize the revenue e∈B∩E T p e collected by the leader. We introduce a reformulation and a Branch-and-cut-and-price algorithm for StackMST. The reformulation is obtained after applying KKT optimality conditions to a StackMST non-compact Bilevel Linear Programming formulation and is strengthened with a partial rank-1 RLT and with valid inequalities from the literature. We also implemented a Branch-and-cut algorithm for an extended formulation derived from another in the literature. A preliminary computational study comparing both methods is also presented.