Summary: Let \(G = (V, E)\) be a simple graph with vertex set \(V\) and edge set \(E\). A Roman dominating function on a graph \(G\) is a function \(f : V(G) \rightarrow \{0, 1, 2 \}\) 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 independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function \(f\) is the value \(f(V(G)) = \sum_{u \in V(G)} f(u)\). The independent Roman domination number of \(G\), denoted by \(i_R(G)\), is the minimum weight of an independent Roman dominating function on \(G\). A subset \(S\) of \(V\) is a 2-independent set of \(G\) if every vertex of \(S\) has at most one neighbor in \(S\). The maximum cardinality of a 2-independent set of \(G\) is the 2-independence number \(\beta_2(G)\). These two parameters are incomparable in general, however, we show that for any tree \(T\), \(\beta_2(T) \geq i_R(T)\) and we characterize all trees attaining the equality.