Why is TREE(3) finite? If the bottom row here is A B C D E F, trees A to D contain each other, only differing in the number of offshoots from the root. Tree F shows that you can add as many offshoots as you want. If these are all valid, why can't we just keep adding offshoots to infinity?

Why is TREE(3) finite? If the bottom row here is A B C D E F, trees A to D contain each other, only differing in the number of offshoots from the root. Tree F shows that you can add as many offshoots as you want. If these are all valid, why can't we just keep adding offshoots to infinity?