The number of total dominating sets in binary trees

Abstract

An (unrooted) binary tree is a tree in which every internal vertex has degree \(3\). In this paper, we determine the minimum and maximum number of total dominating sets in binary trees of a given order. The corresponding extremal binary trees are characterized as well. The minimum is always attained by the binary caterpillar, while the binary trees that attain the maximum are only unique when the number of vertices is not divisible by~\(4\). Moreover, we obtain a lower bound on the number of total dominating sets for \(d\)-ary trees and characterize the extremal trees as well.