Posted: 05/07/2014 at 09:50
The big thick roots anchor the tree. Damage those and you will get instability and a potentially damaging and costly tree fall in a storm. The fine, fibrous ones feed and water the tree. Reducing those will reduce the tree's vigour but only temporarily as they regrow quickly.
Having bought a garden with trees and neighbours it behooves you to work with, rather than against them, especially those belonging to neighbours. For trees in your son's garden, get a tree surgeon to advise whether the whole tree or part of the tree or some of the roots can be safely removed and get them to do the job properly. They and their work should be fully insured and guaranteed. They can also advise you on whether or not the neighbours' trees are too close to buildings or drains and whether or not pruning or removal is advisable. Don't do anything without advice and without checking tree preservation orders and local council policy on trees or you could find yourself with bigger problems down the line. Don't do anything without first talking to your son's neighbours. It's a basic courtesy and he will have to live with them for years so it's best to stay friendly.
As Dove says, you can also use the trees as a design feature and make a shady woodland corner that future grandchildren will love to play in. There are lots of plants that will thrive their and make it beautiful. Have alook here for some ideas - http://www.bethchatto.co.uk/gallery/woodland-garden.htm