Abstract: A subset D ⊆ V (G) in a graph G is a dominating set if every vertex in V (G) \ D is adjacent to at least one vertex of S. A subset S ⊆ V (G) in a graph G is a 2-independent set if ∆(G[S])<2. The 2-independence number α₂(G) is the order of a largest 2-independent set in G. Further, a subset D ⊆ V (G) in a graph G is a 2-independent dominating set if D is both dominating and 2-independent. The 2-independent domination number i²(G) is the order of a smallest 2-independent dominating set in G. In this paper, we characterize all trees T of order n with i²(T) = n/2. Moreover, we prove that for any tree T of order n≥2, i²(T)≤2/3α₂(T), and this bound is sharp.

Key words: Domination, Independent domination, 2-Independent domination, 2-Independence, Trees