As I described, I'd need way fewer, as the optimal computer counter-move would already be included in the next board.
So if you placed your X in the top left field in the starter image, the link would directly go to a field with the X in the top left, the O in the center position, and links in all the remaining seven positions. And of course the pre-calculation will eliminate some of the boards already, e.g. if the player or computer already won after the third move, where placing a fourth will not make sense.