Upper bound for the number of maximal dissociation sets in trees

Let $G$ be a simple graph. A dissociation set of $G$ 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. We introduce the notation $\mathrm{\Phi }(G)$ to represent the number of maximal dissociation sets in $G$. This study focuses on trees, specifically showing that for any tree $T$ of order $n\ge 4$, the following inequality holds: $$\mathrm{\Phi }(T)\le 3^{\frac{n-1}{3}}+\frac{n-1}{3}\mathrm{.}$$ We also identify the extremal tree that attains this upper bound. Additionally, to establish the upper bound on the number of maximal dissociation sets in trees of order $n$, we also determine the second largest number of maximal dissociation sets in forests of order $n$.