The maximum number of maximal dissociation sets in trees

Let $G$ be a simple graph. A dissociation set of $G$ proposed by Yannakakis in $1981$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most $1$. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. In $2025$, Wang et al.\cite{ZiyuanWang} established that for any tree $T$ of order $n\ge 4$, the number of maximal dissociation sets in $T$ is at most $3^{\frac{n-1}{3}}+\frac{n-1}{3}$ and characterized the extremal trees attaining the upper bound. They also proposed a conjecture about the upper bound of the maximal dissociation set. In this paper, we consider this conjecture and show that the maximum number of maximal dissociation sets in a tree of order $n(n\ge 3)$ is $g(n)$, where $$g(n)=\{\begin{array}{ll}n,& n=3,4,5,6,\\ 3^{\frac{n-1}{3}}+\frac{n-1}{3},& n\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}3),n\ge 7,\\ 4\cdot 3^{\frac{n-5}{3}}+n-5,& n\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3),n\ge 8,\\ 16\cdot 3^{\frac{n-9}{3}}+3n-25,& n\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}3),n\ge 12\text{ and }n\mathrm{\ne }21,\\ 19,& n=9,\\ 1349,& n=21.\end{array}$$ We also characterize the extremal trees with the maximum number of maximal dissociation sets.