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 $\Phi \left(G\right)$ 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:$\Phi \left(T\right)\le 3^{\frac{n-1}{3}}+\frac{n-1}{3}.$ We also identify extremal trees that attain 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.