For a subset of vertices \(S\) in a graph \(G\), if \(v \in S\) and \(w \in V-S\), then the vertex \(w\) is an \(external\; private\; neighbor\; of \;v\) (with respect to \(S\)) if the only neighbor of \(w\) in \(S\) is \(v\). A dominating set \(S\) is a private dominating set if each \(v \in S\) has an external private neighbor. Bollébas and Cockayne (Graph theoretic parameters concerning domination, independence and irredundance. J. Graph Theory \(3 (1979) 241-250)\) showed that every graph without isolated vertices has a minimum dominating set which is also a private dominating set. We define a graph \(G\) to be a \(private\; domination\; graph\) if every minimum dominating set of \(G\) is a private dominating set. We give a constructive characterization of private domination trees.