The battle tower is basically a scale down version of actual competitive Pokemon. In gen three and four, the tower makes you fight in seven consecutive battles with no breaks in between. Your mons are healed up at the end of every battle and after the seventh battle, you are given the option to save. If you lose in one of these group of seven battles, then you start over. After a certain number of battles, then you get to challenge the boss of the facility
In gen eight, it seems to be an elo ranking of some kind. If you lose, you don't restart from the beginning but instead your rank may decrease
The battle tree is mostly the same as the tower in three and four, the only difference is that after every battle, you are allowed to have the option to put your challenge on hold. This means that you can save in between of each battle. There are also some fancy options like scouting a trainer so you can team up with them in multi battles. I would say the best part happens in every tenth battle where you get to fight a random boss while in the fiftieth, you fight a certain boss
The important thing to note is that all of these battle facilities are in set mode. Anyway, here are some usful links if you want more information
https://bulbapedia.bulbagarden.net/wiki/Battle_Tree
https://bulbapedia.bulbagarden.net/wiki/Battle_Tower_(Generation_III)
https://bulbapedia.bulbagarden.net/wiki/Battle_Tower_(Generation_IV)
https://bulbapedia.bulbagarden.net/wiki/Battle_Tower_(Generation_VIII)
