Neighbor tree law varies by state - and courts often make rulings that establish new precedents. The basic answer seems to be that you can get rid of the roots, but you can't kill the tree. This holds true for overhanging limbs as well - if the limbs or roots are on your property you can prune them, but you can't do it in such a way that kills the tree, nor can you trespass on the tree owner's property to do the work. This is more of an issue for pruning, as that means you may have to cut a branch at the property line rather than at an appropriate branch union.

As a practical matter - you can cut many tree roots without injuring the tree - the key is the size of root you're cutting and the distance the cut is from the base of the tree. Any larger than about 4 inch diameter roots, and any closer than 3 times the diameter of the tree (so for example, if the tree is 2 feet in diameter, you shouldn't cut roots within 6 feet of the base of the tree) are the rules of thumb that we use as arborists. Also, when cutting roots, they should be cut off cleanly with a saw.

Lastly, regarding the point about your neighbor being liable for damages, that is a legal question, not a tree question, so I'd urge you to contact an attorney if that is an issue.

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