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 γtr(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 γtr(T ) ≥ d(n + 2)/2e. Moreover, we show that if T is a tree of order n ≡ 0 mod 4, then γtr(T ) ≥ d 2 e + 1. We then constructively characterize the extremal trees T of order n achieving these lower bounds.