On the maximum number of minimum dominating sets in forests

Fricke, Hedetniemi, Hedetniemi, and Hutson asked whether every tree with domination number $\gamma$ has at most $2^{\gamma }$ minimum dominating sets. Bien gave a counterexample, which allows to construct forests with domination number $\gamma$ and ${2.0598}^{\gamma }$ minimum dominating sets. We show that every forest with domination number $\gamma$ has at most ${2.4606}^{\gamma }$ minimum dominating sets, and that every tree with independence number $\alpha$ has at most $2^{\alpha -1}+1$ maximum independent sets.