Trees with equal average domination and independent domination numbers

For a vertex D of a graph G = (V,E), the domination number γ_v(G) of G relative to v is the minimum cardinality of a dominating set in G that contains v. The average domination number of G is γ_av(G) = 1/|V| ∑_v∈V(G). The independent domination number i_v(G) of G relative to v is the minimum cardinality of a maximal independent set in G that contains v. The average independent domination number of G is i_av(G) = 1/|V| ∑_v∈V i_v(G). In this paper, we show that a tree T satisfies γ_av(T) = i_av(T) if and only if A(T) = Φ or each vertex of A(T) has degree 2 in T, where A(T) is the set of vertices of T that are contained in all its minimum dominating sets.