Trees, including minimum spanning trees (MSTs), are commonly used in phylogenetic studies. But, for the research community, it may be unclear that the presented tree is just a hypothesis, chosen from among many possible alternatives. In this scenario, it is important to quantify our confidence in both the trees and the branches/edges included in such trees. In this paper, we address this problem for MSTs by introducing a new edge betweenness metric for undirected and weighted graphs. This spanning edge betweenness metric is defined as the fraction of equivalent MSTs where a given edge is present. The metric provides a per edge statistic that is similar to that of the bootstrap approach frequently used in phylogenetics to support the grouping of taxa. We provide methods for the exact computation of this metric based on the well known Kirchhoff's matrix tree theorem. Moreover, we implement and make available a module for the PHYLOViZ software and evaluate the proposed metric concerning both effectiveness and computational performance. Analysis of trees generated using multilocus sequence typing data (MLST) and the goeBURST algorithm revealed that the space of possible MSTs in real data sets is extremely large. Selection of the edge to be represented using bootstrap could lead to unreliable results since alternative edges are present in the same fraction of equivalent MSTs. The choice of the MST to be presented, results from criteria implemented in the algorithm that must be based in biologically plausible models.