  For a simple graph $G(V,E)$, the zeroth-order general Randi\' c index is
defined as $^0R_{\alpha}(G)=\sum_{v\in V(G)}d(v)^{\alpha}$, where $d(v)$ is the
degree of the vertex $v$ and $\alpha\ne0$ is a real number. The $k$-generalized
quasi-tree is a connected graph $G$ with a subset $V_k\subset V(G)$, where
$|V_k|=k$ such that $G-V_k$ is a tree, but for any subset $V_{k-1}\subset V(G)$
with cardinality $k-1$, $G-V_{k-1}$ is not a tree. In this paper, we
characterize the extremal $k$-generalized quasi trees with the minimum and
maximum values of the zeroth-order general Randi\' c index for $\alpha\neq 0$.
