Domination number, independent domination number and k-independence number in trees

For any graph $G$, let $\gamma \left(G\right)$ and $i\left(G\right)$ denote the domination number and the independent domination number of $G$, respectively. For any positive integer $k$, a subset $S$ of vertices in a graph $G$ is said to be a $k$-independent set of $G$ if $G\left[S\right]$ has maximum degree less than $k$. The $k$-independence number of $G$, denoted by ${\alpha }_k\left(G\right)$, is the maximum cardinality of a $k$-independent set of $G$. Let $T$ be any tree with $n\ge 2$ vertices. Dehgardi et al. proved that $i\left(T\right)\le \frac{3}{4}{\alpha }_2\left(T\right)$ and $\gamma \left(T\right)+i\left(T\right)\le \frac{4}{3}{\alpha }_2\left(T\right)$. Later, Zhang and Wu extended the former result of Dehgardi et al. by showing that $i\left(T\right)\le \frac{k+1}{2k}{\alpha }_k\left(T\right)$, and conjectured that the latter one can also be generalized to $\gamma \left(T\right)+i\left(T\right)\le \frac{2k}{2k-1}{\alpha }_k\left(T\right)$. In this paper, we prove this conjecture, and moreover, we characterize all extremal trees for which the equality holds.