SAMS Subject Classification Number: 6

A subset \(D\) of the vertex set \(V(G)\) is a dominating set of graph \(G\) if every vertex of \(V(G) \setminus D\) has a neighbor in \(D\). The set \(D\) is a total dominating set of \(G\) if every vertex of \(G\) has a neighbor in \(D\). A tree is called a binary tree if all it’s internal vertices (i.e., non leaves) are exactly of degree three. Here, we determine the classes of binary trees that has the minimum number of ordinary and total dominating sets. More generally, we extend our results to \(d\)-ary trees.