A characterization of even order trees with domination number half their order minus one

We consider finite graphs G with vertex set V (G). A subset D ⊆ V (G) is a dominating set of the graph G, if every vertex v ∈ V (G) − D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality among the dominating sets of G. In this note, we characterize the trees T with an even number of vertices such that γ(T ) = |V (T )| − 2