Given a group G, we define the power graph P(G) as follows: the vertices are the elements of G and two vertices x and y are joined by an edge if hxi hyi or hyi hxi. 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 < 53. Finally, we show that the alternating group A5 is uniquely determined by tree-number of its power graph among all finite simple groups
