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.