When "Old Scratch," the devil, shows Tom Walker a tree that seemed to be healthy and vigorous but had been scored by an ax to reveal a rotten core, Deacon Peabody's name is cut into it. "Old Scratch" implies that Deacon Peabody will be damned unless he examines his own sins.
"Deacon Peabody be d——d," said the stranger, "as I flatter myself he will be, if he does not look more to his own sins and less to his neighbor's..."
During that same conversation, Tom realizes that the tree he is sitting on, which has just fallen, is inscribed with the name of Absalom Crowninshield. Tom notices that other trees in the swamp have names carved into them, and they all appear to be scored by an ax.
When Tom returns home,
"The first news his wife had to tell him was the sudden death of Absalom Crowninshield the rich buccaneer. It was announced in the papers with the usual flourish, that "a great man had fallen in Israel."
The implication is that Crowninshield has been literally "cut down" by the devil, because his crime of buccaneering is widely acknowledged. Presumably, his life and soul have been claimed by Old Scratch because Crowninshield made the same deal with him that he offers Tom Walker: temporal riches in exchange for his immortal soul.
The words "a great man has fallen in Israel" have a double meaning: Crowninshield is dead, but he also experienced a moral fall.
Symbolically, then, the trees represent men who are at grave risk of death and damnation unless they mend their ways. Swamps have long been employed by writers as a symbol for a place of moral murkiness, so it is by design that Irving places the trees here.