Alligators cannot climb trees. They live in the water and come on to land for short periods of time to hunt, and sun themselves. Nothing in trees is of interest to them. They wait for the birds, ducks and other tree dwellers to come to them.

An alligator can climb a tree. It is very uncommon but it has been done. If you are being chased by an alligator and you think it is safe to climb a tree your probably wrong, if the alligator wants you bad enough he will climb.