Assume \(T\) is a tree, and let \(u\) and \(v\) be distinct vertices (if \(T\) only has one vertex, then the conclusion is satisfied automatically). We must show two things to show that there is a unique path between \(u\) and \(v\text{:}\) that there is a path, and that there is not more than one path. The first of these is automatic, since \(T\) is a tree, it is connected, so there is a path between any pair of vertices.
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