Summary: Let \(G=(V, E)\) be a simple graph with vertex set \(V\) and edge set \(E\). A {em mixed Roman dominating function} (MRDF) of \(G\) is a function \(f:V\cup E\rightarrow \{0,1,2\}\) satisfying the condition that every element \(x\in V\cup E\) for which \(f(x)=0\) is adjacent or incident to at least one element \(y\in V\cup E\) for which \(f(y)=2\). The weight of an MRDF \(f\) is \(\sum _{x\in V\cup E} f(x)\). The mixed Roman domination number \(\gamma^*_R(G)\) of \(G\) is the minimum weight among all mixed Roman dominating functions of \(G\). 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 minimum 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 if \(T\) is a tree, then \(\frac{4}{3}\beta_2(T)\geq \gamma^*_R(T)\) and we characterize all trees attaining the equality.