Sampathkumar and Pushpa Latha (see \({[3]}\)) conjectured that the independent domination number, \(i(T)\), of a tree \(T\) is less than or equal to its weak domination number, \(\gamma_w(T)\). We show that this conjecture is true, prove that \(\gamma_w(T) \leq \beta(T)\) for a tree \(T\), exhibit an infinite class of trees in which the differences \( \gamma_w-i \) and \(\beta – \gamma_w\) can be made arbitrarily large, and show that the decision problem corresponding to the computation of \(\gamma(G)\) is \(NP\)-complete, even for bipartite graphs. Lastly, we provide a linear algorithm to compute \(\gamma_w(T)\) for a tree \(T\).