given a vertex v, I want compute all shortest paths starting at v, i.e.,
I want a list of edge or vertex lists, or some kind of DAG. Of course
this can be easily solved by Dijkstra's Algorithm, but Graph-Tools
doesn't seem to provide the right interface for this problem.

The leaves are just the nodes that are visited last by the BFS/Dijkstra,
i.e. the ones that are furthest away from the source. Are you sure this
is what you want? In any case, the answer is no, you have to find them
in the tree.

Note however that the ordering of the nodes in this tree are the same as
in the original graph. So if you want to traverse the
successor/predecessor of any particular node v in the original graph,
you can lookup t.vertex(v) in time O(1), where t is the predecessor
tree.