Trees with distinguishing index equal distinguishing number plus one

The distinguishing number (index)D(G) (D′(G)) of a graphGis theleast integerdsuch thatGhas an vertex (edge) labeling withdlabels thatis preserved only by the trivial automorphism. It is known that for everygraphGwe haveD′(G)≤D(G) + 1. In this note we characterize finite treesfor which this inequality is sharp. We also show that ifGis a connectedunicyclic graph, thenD′(G) =D(G)