A set of vertices in a graph is a total dominating set if every vertex of is adjacent to some vertex in . The total domination number is the minimum cardinality of a total dominating set in . The annihilation number is the largest integer such that the sum of the first terms of the non-decreasing degree sequence of is at most the number of edges in . In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let be a tree of order . We show that , and we characterize the extremal trees achieving equality in this bound.