Tom encounters the Devil deep in a swamp, dressed more or less as a lumberjack. When he challenges the Devil (while unaware of his true identity) that the land belongs to Deacon Peabody, the Devil directs Tom's attention to a nearby tree;
Tom looked in the direction that the stranger pointed, and beheld one of the greattrees, fair and flourishing without, but rotten at the core, and saw that it had beennearly hewn through, so that the first high wind was likely to blow it down. On thebark of the tree was scored the name of Deacon Peabody
The rotting represents the sins of these men; great without, but corrupt within. The Devil's chopping has laid bare their failings. When it is revealed that the name borne on another fallen tree, that of Crowninshield, has actually died, the meaning of the trees becomes more complicated; perhaps the trees are meant to actually represent the men themselves, and not merely serve as symbols for them. Thus, when the tree dies, so too does the man; in this manner, the trees might be regarded as an analogy to the Thread of Life measured and cut by the Fates of Grecian mythology, with the Devil acting as the Fates.
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.