Mixed Roman domination and 2-independence in trees

‎‎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∪E→{0,1,2} satisfying the condition that every element xinV∪E for which f(x)=0 is adjacent‎ ‎or incident to at least one element y∈V∪E for which f(y)=2‎. ‎The weight of an‎ ‎MRDF f is ∑x∈V∪Ef(x)‎. ‎The mixed Roman domination number γ∗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 β2(G)‎. ‎These two parameters are incomparable in general‎, ‎however‎, ‎we show that if T is a tree‎, ‎then 43β2(T)≥γ∗R(T) and we characterize all trees attaining the equality‎.