Many works within robust combinatorial optimisation consider interval-valued costs or constraints. While most of these works focus on finding unique solutions such as minimax ones, a few consider the problem of characterising a set of non-dominated optimal solutions. This paper is situated within this line of work, and consider the problem of exactly enumerating the set of non-dominated spanning trees under interval-valued costs. We show in particular that each tree in this set can be obtained through a polynomial procedure, and provide an efficient algorithm to achieve the enumeration.