A minimum spanning tree is a tree. It is different from other trees in that it minimizes the total of the "weights" attached to the edges. Depending on what the graph looks like, there may be more than one minimum spanning tree. In a graph where all the edges have the same weight, every tree is a minimum spanning tree. If all the edges have different weights (that is: there are no two edges with the same weight), there is exactly one minimal spanning tree. 