Rip a sheet of aluminum foil into long strips and tie these pieces to the tree’s lower limbs. The aluminum foil strips wave easily even in a slight breeze, and the reflecting light should discourages woodpeckers. You will need to experiment with quantities to determine how many are needed to deter the birds.
Hang a windsock or two from limbs as further discouragement.
Mount a plastic owl on a tree limb. The owl must be moved periodically for it to be effective.
Play a CD of woodpecker distress calls that are followed by predator calls, such as those made by a sharp-shinned hawk. These CDs are available at birding retailers. Keep the player under the tree and play the CD anytime woodpeckers roost or peck on the tree.