If you own a tree that is not in a Conservation Area, is not designated with a Tree Preservation Order, and is not part of any planned development that would require planning permission, then you can undertake any work to that tree.
If you live next to a property with a tree, under common law you may cut that tree back to your boundaries, but you cannot enter the property of the landowner upon whose land the tree is situated without their express permission and you must offer the arisings to the tree owner.