Should we expect the pole in the free style indoor cat tree to rotate at all?

A slight rotational movement can't be prevented completely with a set-up that has got one vertical pole only. To decrease the movement of the pole we can advise adding a wall brace to your set-up, this will not stop the movement completely but certainly will help to reduce it. 

Another option is to go for a two pole set up and connect the two poles with a horizontal pole, a ramp or a bridge. This connection will stop the rotational movement.