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.
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.