Well first you need a pokemon that can use cut. Then stand in front of the tree and press the A(talk/action) button. There will be a question asking if you would like to cut the tree. Another way of doing this is to stand in front of the tree, open up your menu, clicking on the pokemon that can use cut and press cut.
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You can't, you need the HM CUT.
You have to get cut, which is given to you by someone in the game. You then have to beat a gym leader and get a badge to allow the Pokemon to use the move.
If there is a pokemon in your party that can use Cut and you have the second badge then face a tree that can be Cut down press A on it then press A on YES to cut the tree alternatively you can open the pokemon screen and select your pokemon and press A on Cut.
No
First teach your Pokemon cut then you must beat the 2nd gym leader now press Start and press A on Pokemon your Pokemon will be displayed choose the Pokemon you taught Cut to and press A on it a menu will show up with diffrent commands choose Cut now your Pokemon will use cut.