Number of maximal 2-component independent sets in forests

Let $G=(V(G),E(G))$ be a graph. For a positive integer $k$, we call $S\subseteq V(G)$ a $k$-component independent set of $G$ if each component of $G[S]$ has order at most $k$. Moreover, $S$ is maximal if there does not exist a $k$-component independent set $S^{\prime }$ of $G$ such that $S\subseteq S^{\prime }$ and $|S|<|S^{\prime }|$. A maximal $k$-component independent set of a graph $G$ is denoted briefly by Mk-CIS. We use $t_k(G)$ to denote the number of Mk-CISs of a graph $G$. In this paper, we show that for a forest $G$ of order $n$,

$$t_2(G)\le \{\begin{array}{ll}{3}^{\frac{n}{3}},& \text{if}n\equiv 0(mod3)\text{and}n\ge 3,\\ 4\cdot 3^{\frac{n-4}{3}},& \text{if}n\equiv 1(mod3)\text{and}n\ge 4,\\ 5,& \text{if}n=5,\\ 4^2\cdot 3^{\frac{n-8}{3}},& \text{if}n\equiv 2(mod3)\text{and}n\ge 8,\end{array}$$

with equality if and only if $G\cong F_n$, where

$$F_n\cong \{\begin{array}{ll}\frac{n}{3}{P}_3,& \text{if}n\equiv 0(mod3)\text{and}n\ge 3,\\ \frac{n-4}{3}{P}_3\cup K_{1,3},& \text{if}n\equiv 1(mod3)\text{and}n\ge 4,\\ K_{1,4},& \text{if}n=5,\\ \frac{n-8}{3}{P}_3\cup 2K_{1,3},& \text{if}n\equiv 2(mod3)\text{and}n\ge 8.\end{array}$$