I had one about 12ft high in (all over...) my tiny front garden. It got cut down and main roots removed by a bloke with a pickaxe before I dug around with a fork and pulled out as much as I could of the left over bits. We removed it to a foot under the ground! It has not returned. If your neighbour's trees are well established they should survive the accidental removal of a few bits of root.

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You are a child of the universe,
no less than the trees and the stars;
you have a right to be here.
Max Ehrmann, Desiderata

blog: http://allyheebiejeebie.blogspot.com/ and my (basic!) page: http://www.allythegardener.co.uk/