Trees with equal 2-domination and 2-independence numbers

Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V-S is dominated at least 2 times, and S 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-dominating set a of G is the 2-domination number γ₂(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β₂(G). Fink and Jacobson proved that γ₂(G) ≤ β₂(G) for every graph G. In this paper we provide a constructive characterization of trees with equal 2-domination and 2-independence numbers.