One of the easiest ways to tell the difference between a Rat Snake and a Copperhead is to look at its eyes. A rat snake has eyes with a round pupil whereas the Copperhead has eyes that resemble those of a cat. I do not believe either of them have red bellies. A copperhead also has a yellow tail during its juvenielle stage. Also Rat Snakes have no problem climbing up trees but you will not find a Copperhead climbing a tree. It sounds like what you saw could be a Red-Bellied Water Snake or a Red-Bellied Snake. Do a quick google search and see if either resembles what you saw!