ANONYMOUS wrote: The defining property of a spanning tree is that it is a tree that contains all vertices in the graph. The defining property of a shortest path tree is that every path starting from the root of the tree is a shortest path in the graph (typically to everything the root can reach). These are independent properties, and it is possible for a subtree of a graph to be one but not the other, both, or neither. So no, not all shortest path trees are spanning trees. In particular, consider the case with a disconnected graph where there is not a path to every vertex from the root. If the root is able to reach every other vertex, then yes, its shortest path tree will typically be a spanning tree.