  The zeroth-order general Randi\'c index of graph $G=(V_G,E_G)$, denoted by
$^0R_{\alpha}(G)$, is the sum of items $(d_{v})^{\alpha}$ over all vertices
$v\in V_G$, where $\alpha$ is a pertinently chosen real number. In this paper,
we obtain the sharp upper and lower bounds on $^0R_{\alpha}$ of trees with a
domination number $\gamma$, in intervals $\alpha\in(-\infty,0)\cup(1,\infty)$
and $\alpha\in(0,1)$, respectively. The corresponding extremal graphs of these
bounds are also characterized.
