Spanning trees implies isomorphism?

I'm having trouble solving this question:

Prove or disprove the following statement: Given a graph G, if T and U are spanning trees of G, then T and U are isomorphic.

I know that two graphs are isomorphic if they have an "edge-preserving bijection." I also know that a spanning tree of a graph G is a connected graph that can be defined as a maximal set of edges of G that contains no cycle.

How would I go about proving or disproving?

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Or of $K_4$, the complete graph on 4 vertices. –  Gerry Myerson Nov 14 '12 at 4:39