This paper investigates the Bi-objective Spanning Tree Problem (BSTP), an extension of the fundamental Minimum Spanning Tree problem with diverse applications. In this problem, each edge of a given graph is associated with two distinct weights, and the objective is to find a spanning tree that simultaneously minimizes both total weights. As a specific case of bi-objective optimization, we may be interested in enumerating all the Pareto-optimal solutions of the BSTP. However, as the BSTP has been shown to be NP-hard, all the existing approaches to enumerate efficient solutions cannot achieve a polynomial CPU time. In this paper, we propose a novel approach for finding preferred efficient solutions where the worst of the ratios of the two objective values to their maximum possible values, respectively, is minimized. This approach is based on the Kalai-Smorodinsky (KS) solution, a well-known concept in cooperative game theory. More precisely, we consider an extension of the KS solution concept to the non-convex case that can be found by optimizing convex combinations of two objectives. We first characterize the properties of such KS solution(s) for the BSTP. Next, we present a weakly polynomial-time algorithm for finding the KS solution(s) for the BSTP. Finally, we showcase the computational results in some instances and discuss the results. Beyond the particular case of the BSTP, this paper offers an efficient and explainable approach for solving bi-objective combinatorial optimization problems.