I have a neighbor outside trying to get her parrot to come down out of a tree. She's apparently been out there for a while trying to coax the bird down.
Are there any proven methods for re-capturing an escaped pet bird without harming the bird?
Put the bird's cage outside, as close as possible to the tree where the bird is, and put some food and water in it, preferably the kind of food that the bird enjoys the most. Leave the cage door open, and wait. The bird will probably be hungry after some time outside, especially if it doesn't know how to forage for itself, and it may also be scared from being out of its usual environment, and see the cage as safety.
Calling the bird's name might also work, but it sounds like your neighbor has already been doing this.