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A set \(D\) of vertices in a graph \(G = (V, E)\) is a locating-dominating set if for every two vertices \(u, v\) in \(V \setminus D\), the sets \(N(u) \cap D\) and \(N(v) \cap D\) are non-empty and different. We establish two equivalent conditions for trees with unique minimum locating-dominating sets.