12 gauge, full choke, 7 1/2 shot should do the trick. Shoot nearly straight up, your shot should still be in a cluster as you hit the branch, but will spread out after that. It's small enough it can't do any damage on the way down. Warn your neighbors you're testing fireworks for the 4th of July so they don't call the police.

I knew somebody who did this to remove mistletoe from his trees. He bought a new shotgun and a case of ammo for about half the price the tree trimming firm quoted him.