If you are doing self-play and building the tree exactly the same for both players there won't be any bias inherent in the tree - you can re-use it for both players. But, if the players build the MCTS tree in a way that is specific to a particular player, then you'll need to rebuild the tree. In this case you'd need to keep two trees, one for each player, and each player could re-use their own tree but nothing else.
Some things to analyze if you're trying to figure this out:
- Does the game have hidden information? (Something one player knows that the other player doesn't.) In this case you can't re-use the tree because you'd leak private information to the other player.
- Do your playouts depend on the player at the root of the MCTS tree?
- Does you have any policies for pruning moves from either player that aren't applied symmetrically?
- Do you evaluate states in a way that is not symmetric between players?
- Do you perform any randomization differently for the players?
If none of these are true, you can likely re-use the tree.