Summary: A Roman dominating function (RDF) on a graph \(G\) is a function \(f : V (G) \to \{0, 1, 2\}\) satisfying the condition that every vertex \(u\) for which \(f(u) = 0\) is adjacent to at least one vertex \(v\) for which \(f(v) = 2\). A Roman dominating function \(f\) is called an outer-independent Roman dominating function (OIRDF) on \(G\) if the set \(\{v\in V\mid f(v)=0\}\) is independent. The (outer-independent) Roman domination number \(\gamma_{\mathrm{R}}(G) (\gamma_{\mathrm{oiR}}(G))\) is the minimum weight of an RDF (OIRDF) on \(G\). Clearly, for any graph \(G\), \(\gamma_{\mathrm{R}}(G)\le \gamma_{\mathrm{oiR}}(G)\). In this paper, we provide a constructive characterization of trees \(T\) with \(\gamma_{\mathrm{R}}(T)=\gamma_{\mathrm{oiR}}(T)\).