A subset D of vertices in a graph G is a dominating set if every vertex in \(V(G)\setminus D\) is adjacent to at least one vertex in D. For any positive integer k, a subset S of vertices in G is a k-independent set if G[S] has maximum degree less than k. The k-independence number of G, denoted by \(\alpha _k(G)\) , is the maximum cardinality of a k-independent set in G. A subset I of vertices in G is a k-independent dominating set if I is both k-independent and dominating. The k-independent domination number of G, denoted by \(i_k(G)\) , is the minimum cardinality of a k-independent domination set in G. Recently, Zhang and Wu (J Oper Res Soc China 12:485–494, 2024) showed that \(i_2(T)\leqslant \frac{2}{3}\alpha _2(T)\) for any nontrivial tree T, and this bound is sharp. In this paper, we give a complete characterization of all trees attaining this bound, which resolves a problem proposed by Zhang and Wu. Moreover, we further prove that \(i_3(T)\leqslant \frac{3}{5}\alpha _3(T)\) for any nontrivial tree T, and characterize all extremal trees for which the equality holds.
