We study the concept of strong equality of domination parameters. Let P 1 and P 2 be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P 2 also has property P 1 . Let ψ 1 (G) and ψ 2 (G), respectively, denote the minimum cardinalities of sets with properties P 1 and P 2 , respectively. Then ψ 1 (G)=<ψ 2 (G). If ψ 1 (G)=ψ 2 (G) and every ψ 1 (G)-set is also a ψ 2 (G)-set, then we say ψ 1 (G) strongly equals ψ 2 (G), written ψ 1 (G)=ψ 2 (G). We provide a constructive characterization of the trees T such that γ(T)=i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ t (T), where γ t (T) denotes the total domination number of T, is also presented.