  A \emph{perfect Roman dominating function} (PRDF) on a graph $G = (V, E)$ is
a function $f : V \rightarrow \{0, 1, 2\}$ satisfying the condition that every
vertex $u$ for which $f(u) = 0$ is adjacent to exactly one vertex $v$ for which
$f(v) = 2$. The weight of a PRDF is the value $w(f) = \sum_{u \in V}f(u)$. The
minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman
domination number $\gamma_R^p(G)$} of $G$. A graph $G$ is perfect Roman
domination domination stable if the perfect Roman domination number of $G$
remains unchanged under the removal of any vertex. In this paper, we
characterize all trees that are perfect Roman domination stable.
