On the number of minimum total dominating sets in trees

The minimum total dominating set (MTDS) of a graph is a vertex subset $D$ of minimum cardinality such that every vertex of the graph is adjacent to at least one vertex of $D.$ In this paper we obtain the sharp upper bound for the number of MTDS in the class of $n$-vertex $2$-caterpillars. We also show that for all $n \geq 1$ every $n$-vertex tree has less than $(\sqrt{2})^n$ MTDS. Illustr. 5, bibliogr. 6.