We analyze in details the biobjective minimum spanning trees (MSTs) problem in the case in which the first objective function is a linear one (minimum length) and the second objective function is a non-linear bottleneck (minimum risk). We propose an exact method that constructs the complete Pareto front of the problem with computational complexity \(O(\beta (m + \alpha n \lg {n}))\) , where n and m are the sizes of the input network, \(\alpha \) is a relatively small parameter, and \(\beta \) depends on the number of MST in the original network and its restricted versions that are calculated on each iteration. As a part of our solution, we propose an efficient exact algorithm that constructs all MSTs for the single-objective linear problem that is based on a modification of the Prim’s algorithm. The proposed modification relies on a greedy property that is proved mathematically. We provide detailed proofs of the correctness of the proposed algorithms, and we also illustrate them with numerical examples.