In the field next to my house are a number of silver birch trees and willows. They were planted after my house was built in 1969 and at the nearest point are only ten feet away. What would be the position if falling branches damaged my roof, or if the roots undermined my foundations?
You should probably ask the owner of the field to prune the trees to avoid the possibility of falling branches damaging your house. If damage did occur your best bet would be to contact your insurance company; they in turn could attempt to recover the cost of repairs from your neighbour, but in any case, assuming you have kept your insurance premiums up to date, they would be responsible for making good any damage.