The minimum cost spanning tree game (mcst-game) is a well-known model within operations research games that has been widely studied in the literature. In this paper we introduce the multi-criteria version of the mcst-game as a set-valued TU-game. We prove that the extension of Bird's cost allocation rule provides dominance core elements in this game. We also give a family of core solutions that are different from the previous one; these solutions are based on proportional allocations obtained using scalar solutions of the multi-criteria spanning tree problem. Besides, we prove necessary and sufficient conditions ensuring that the preference core of this game is not empty
