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The distinguishing number (index) D(G) (D (G)) of a graph G is the least integer d such that G has an vertex (edge) labeling with d labels that is preserved only by the trivial automorphism. It is known that for every graph G we have D (G) ≤ D(G) + 1. In this note we characterize finite trees for which this inequality is sharp. We also show that if G is a connected unicyclic graph, then D (G) = D(G).doi:10.7151/dmgt.2162 fatcat:wpzwkcroo5fftaqs6zjilvsn2i