  Consider the nearest neighbor graph for the integer lattice Z^d in d
dimensions. For a large finite piece of it, consider choosing a spanning tree
for that piece uniformly among all possible subgraphs that are spanning trees.
As the piece gets larger, this approaches a limiting measure on the set of
spanning graphs for Z^d. This is shown to be a tree if and only if d=<4. In
this case, the tree has only one topological end, i.e. there are no doubly
infinite paths. When d>=5 the spanning forest has infinitely many components
almost surely, with each component having one or two topological ends.
