The good news is it docent look like fin rot. Although fin rot is usually ragged fin looking, fin rot is caused by gram-rod negative bacteria that causes the end of the fins to become whitish and fuzzy, or almost black and burnt looking. Your in luck, because i am dealing with the same thing your Betta seems to have. My Betta Has the same symptoms as yours, i woke up and he was missing some of his fin too. I took some pantyhose and ran it over all the decorations to see if it would snag, and sure enough, one did. I removed the castle (any sharp decorations that the pantyhose snags on signifies a bettas fins tearing) and adding 1 tablespoon of AQUARIUM salt per 5 gallons along with water changes every day to keep him from getting any infection like fin rot. All i have to do now is to wait, as i have been told it will help his fins grow back in a couple of days.