What do the names carved into the trees represent in "The Devil and Tom Walker"?
The names carved into the trees represent those who've sold their souls to the devil. Tom doesn't notice the names on the trees until the black man begins telling him that the land Tom is on belongs to the black man. Also, as proof that Deacon Peabody, the one who Tom believed to be the owner of the land, is not doing well, the black man points to a tree that is large and flourishing at the outer areas, but is rotten at the core. On this tree was written the name, "Deacon Peabody". The black man tells Tom, "Deacon Peabody be d--d...as I flatter myself he will be, if he does not look more to his own sins and less to those of his neighbors." The reader shortly discovers that the identity of the black man is the devil. Furthermore, the black man indicates that as he takes each soul, he cuts down the tree with that man's name on it. He tells Tom that he'll have lots of firewood for the winter.
In addition, the downed trees have names engraved on them that Tom recognizes. These people he remembers as being "wealthy and important" to society in the past, and should have been his first clue not to deal with the Devil, ie black man. His sense of logic is overruled by his sense of greed and ambition.
On that note, it will benefit you to remember that society and reason/logic are two of the classic elements that the Romantics were rebelling against. They much preferred the individual to society as a whole and emotion/imagination to logic and reason.