Why does the woodsman score the trees?
During Tom Walker's first interaction with Old Scratch, he notices that each of the trees encompassing the woodsman's domain is scored with the name of a prominent community member. The first name scored onto a tree that Tom sees is Deacon Peabody. The reader also learns that Deacon Peabody became a wealthy man by "driving shrewd bargains with Indians." Tom also notices that a downed tree has the name Crowninshield scored on its side. The narrator mentions that Absalom Crowninshield had made his wealth through immoral buccaneering. Essentially, Old Scratch scores the names of men who have sold him their souls onto the trees in the forest. The prominent men, whose names are scored onto the trees, sold their souls to the Devil in exchange for wealth. The trees that have been chopped down represent the souls that Old Scratch has collected. When Tom Walker returns to the village, he learns that Absalom Crowninshield has suddenly died. The reader can make the connection that each time the woodsman cuts down a tree with a person's name scored on its side, he takes their life and soul.
The woodsman is the devil, also known as Old Scratch. He has scored the names of people's souls he has collected on the trees. In this story, the forest is portrayed as dark and evil, which is appropriate because it is where Old Scratch can be found. However, the people of the town want to find the money that Kidd the pirate hid under the oak trees, which is what Old Scratch uses to convince people to sell their souls. He promises them the treasure. When Old Scratch is ready to collect the soul, sending the person in to hell, he cuts the tree down and burns it.
All of the trees are more or less scored. Each of the trees represents a person under the influence of the devil, who, as has been stated, is the the woodsman.
The devil gradually cuts down the trees (and the people) under his control. When a tree is completely cut down (like Crowninshield's) that person is "ready for burning," or ready for hell.
The scores in the trees show how much life the people represented have remaining. The deeper the score (or cut) the closer they are to death.