The trees serve as symbols: each one represents the life of a corrupt rich man in the town who has bargained with Old Scratch. When one of the trees is cut down, that person represented also dies.
The metaphor drawn between the life of the corrupt men of the town and the trees themselves is similar to plant symbolism used throughout literature in general: Because both humans and plants are living organisms, authors often symbolize human characteristics through the life/death of any plant.
Roses and other flowers are particularly useful to authors when they are utilizing the above-described device. As roses symbolize love, among other things, the death of a rose can be equated to the death of love. Such is the case with these trees -- the people they represent are directly affected by their deaths.
Each of the trees that Tom Walkers sees on Deacon Peabody's land represented the soul of a prominent and wealthy man in the world. Each of those men has presumably sold his soul to the devil and when the devil decides that it is time for that man to die, he chops down the tree which contains his name. Therefore, when Old Scratch chops down a tree, the man whose name was on the tree dies.