  Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained
dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in
$D$ and to a vertex in $V \setminus D$. The \textit{restrained domination
number}, denoted by $\gamma_r(G)$, is the smallest cardinality of a restrained
dominating set of $G$. A function $f : V \rightarrow \{0, 1, 2\}$ is a
\textit{restrained Italian dominating function} on $G$ if (i) for each vertex
$v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N_G(v)} f(u) \geq 2$,
(ii) the subgraph induced by $\{v \in V \mid f(v)=0 \}$ has no isolated
vertices. The \textit{restrained Italian domination number}, denoted by
$\gamma_{rI}(G)$, is the minimum weight taken over all restrained Italian
dominating functions of $G$. It is known that $\gamma_r(G) \leq \gamma_{rI}(G)
\leq 2\gamma_r(G)$ for any graph $G$. In this paper, we characterize the trees
$T$ for which $\gamma_r(T) = \gamma_{rI}(T)$, and we also characterize the
trees $T$ for which $\gamma_{rI}(T) = 2\gamma_r(T)$.
