The maximum number of maximum generalized 4-independent sets in trees

A generalized $k$-independent set is a set of vertices such that the induced subgraph contains no trees with $k$-vertices, and the generalized $k$-independence number is the cardinality of a maximum $k$-independent set in $G$. Zito proved that the maximum number of maximum generalized 2-independent sets in a tree of order $n$ is $2^{\frac{n-3}{2}}$ if $n$ is odd, and $2^{\frac{n-2}{2}}+1$ if $n$ is even. Tu et al. showed that the maximum number of maximum generalized 3-independent sets in a tree of order $n$ is $3^{\frac{n}{3}-1}+\frac{n}{3}+1$ if $n\equiv 0 \left(\text{mod 3}\right)$, and $3^{\frac{n-1}{3}-1}+1$ if $n\equiv 1 \left(\text{mod 3}\right)$, and $3^{\frac{n-2}{3}-1}$ if $n\equiv 2 \left(\text{mod 3}\right)$ and they characterized all the extremal graphs. Inspired by these two nice results, we establish four structure theorems about maximum generalized $k$-independent sets in a tree for a general integer $k$. As applications, we show that the maximum number of generalized 4-independent sets in a tree of order $n (n\ge 4)$ is