On the maximum number of minimum total dominating sets in forests

We propose the conjecture that every tree with order n at least 2 and total domination number γt has at most ⁿ⁻ γ _2t ^{γ _2t γ 2t} minimum total dominating sets. As a relaxation of this conjecture, we show that every forest F with order n, no isolated vertex, and total domination number γt has at most min (8e ^{γt n −} _{γ 2 t} ^{γ 2 t γ 2t} , (1 + 2) ⁿ −γ ^t , 1.4865 ⁿ⁾ minimum total dominating sets.