Tl;dr: can alligators regrow their tails? Yes. Well, some of it.
According to a team of researchers from Arizona State University and the Louisiana Department of Wildlife and Fisheries, juvenile American alligators have the ability to regrow their tails up to 18% of their total body length.
An interdisciplinary team of scientists used advanced imaging techniques combined with demonstrated methods of studying anatomy and tissue organization to examine the structure of these regrown tails. They found that these new tails were complex structures with a central skeleton composed of cartilage surrounded by connective tissue that was interlaced with blood vessels and nerves. They speculate that regrowing their tails gives the alligators a functional advantage in their murky aquatic habitats.
In terms of what determined the length of regrowth, the team advised variations depended on “sex, age, or environment” due to reptiles being ectotherms and that “tail repair with regrowth in the alligator is a prolonged process.”
Overall, this study of wild-caught, juvenile American alligator tails identifies a distinct pattern of wound repair in mammals while exhibiting features in common with regeneration in lepidosaurs and amphibia.