  Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the
vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an
edge if $\langle x\rangle\subseteq \langle y\rangle$ or $\langle
y\rangle\subseteq \langle x\rangle$. Obviously the power graph of any group is
always connected, because the identity element of the group is adjacent to all
other vertices. In the present paper, among other results, we will find the
number of spanning trees of the power graph associated with specific finite
groups. We also determine, up to isomorphism, the structure of a finite group
$G$ whose power graph has exactly $n$ spanning trees, for $n<5^3$. Finally, we
show that the alternating group $\mathbb{A}_5$ is uniquely determined by
tree-number of its power graph among all finite simple groups.
