Recent years have witnessed a surge of biological interest in the minimum spanning tree (MST) problem for its relevance to automatic model construction using the distances between data points. Despite the increasing use of MST algorithms for this purpose, the goodness-of-fit of an MST to the data is often elusive because no quantitative criteria have been developed to measure it. Motivated by this, we provide a necessary and sufficient condition to ensure that a metric space on n points can be represented by a fully labeled tree on n vertices, and thereby determine when an MST preserves all pairwise distances between points in a finite metric space.