Basic steps:

When n=0, the side length of the square is 2 0 =1,and there is only one grid, in which a lattice tree statue can indeed be vacated.

Inductive hypothesis:

Suppose that when n=k, a lattice tree statue can be left in the center of a square with a side length of 2 k.

Inductive steps:

Consider a square with a side length of 2 (k+ 1) = 2 * 2 k, divide it into four small squares with a side length of 2 k, and a lattice tree statue can be left in the center of each small square (according to the inductive hypothesis). Select the center position of one small square and occupy the center positions of the other three small squares at the same time. Then place a tree statue directly above the four small squares. In this way, a lattice tree statue with a side length of 2 (k+ 1) can be left in the center of the square.

To sum up, according to the principle of mathematical induction, a lattice tree statue can be left in the center of a square with a side length of 2 n.