Total restrained domination in trees

Let G = (V, E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V - S is adjacent to a vertex in V - S. The total restrained domination number of G, denoted by γ_tr (G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then γ_tr (T) ≥ ⌈⌉ fenced(frac(n + 2, 2)). Moreover, we show that if T is a tree of order n ≡ 0 mod 4, then γ_tr (T) ≥ ⌈⌉ fenced(frac(n + 2, 2)) + 1. We then constructively characterize the extremal trees T of order n achieving these lower bounds.