The spanning tree of k3,3 can be divided into two categories: (1)6 vertex paths, and there are 1/2(6 times 3 times 2 times 2)=36 such paths, because 1 vertex has 6 choices, 3 for the second, 2 for the third and 2 for the fourth. (2) The spanning tree contains vertices with degree 3, and such a tree has 3 times 3+6 times 3 times 2=45 trees, because if there are two vertices with degree 3, there is a choice of 3 times 3, and if there are 1 vertices with degree 3, there is a choice of 6 times 3 times 2. So the spanning tree of K3,3 has 8 1 tree.

Alternatively, you can use Caylay recurrence formula to find it.