Could be the fake plants, get some knee high hose (cheap in plastic containers at WalMart) and run them over the plants to see if they snag, if they do it's the plants. Even if not I'd remove them until it heals.
Another thing is glass reflection causing tail biting which makes uneven back fins kind of like a tear. Recent post I did on it & solution:
Then there is tail rot. Sticky at the top describes each & how to treat them: http://www.bettafish.com/showthread.php?t=73332
For torn fins from deco's I'm guessing it's treated like Fin Biting to help them heal and prevent infection.