To show the path is unique, we suppose there are two paths between \(u\) and \(v\text{,}\) and get a contradiction. The two paths might start out the same, but since they are different, there is some first vertex \(u'\) after which the two paths diverge. However, since the two paths both end and \(v\text{,}\) there is some first vertex after \(u'\) that they have in common, call it \(v'\text{.}\) Now consider the two paths from \(u'\) to \(v'\text{.}\) Taken together, these form a cycle, which contradicts our assumption that \(T\) is a tree.
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