As far as I understood, for searching for the best move, yo do not need to search all moves using minimax, you could use alphabeta, that is a kind of cut over the "minimax tree", and just search over it.
So, as far as I understood, you need to store just the "alphabeta tree" that is smaller than the "minimax tree".
And my question if I understood right is:
The number of moves studied that engines use to show to the GUI, are the moves the "minimax tree" have or the moves the "alphabeta tree" has?
I think are the "minimax tree" moves, but not sure.... in fact, as I said, I´m not sure if I am understanding the matter or not.
Thanks.

