A neighbor may remove branches extending over a shared property line onto his or her own property. See, e.g., Levine v. Black, 312 Mass. 242 (1942); Michalson v. Nutting, 275 Mass. 232 at 233-234 (1931). Also, the neighbor has no liability for roots growing into your yard and causing damage. The ability to cut back limbs and roots is limited by Mass. Gen. Laws ch. 87, § 11 that provides: "Whoever wilfully, maliciously or wantonly cuts, destroys or injures a tree, shrub or growth which is not his own, standing for any useful purpose, shall be punished by imprisonment for not more than six months or by a fine of not more than five hundred dollars." You can trim the branches and roots back, but you cannot kill the tree. This is the "Massachusetts Rule."
Massachusetts law does not allow a person to cross or enter a neighbor’s property for these purposes without the neighbor’s consent, nor to remove any branches or other vegetation within the confines of the neighbor’s property. Mass. Gen. Laws ch 242 §7 A party is liable for triple damages for entering the property of another and cutting down trees or branches.
There is also a Massachusetts statute that prohibits cutting, trimming or removing of public shade trees. Mass. Gen. Laws Ch. 87. Under this statute public shade trees are defined as “All trees within a public way or on the boundaries thereof…” Mass. Gen. Laws ch. 87 § 1.
The Massachusetts Rule is followed in most states. However, the Washingtonpost.com
is reporting that Virginia just adopted a new rule: Virginia High Court Breaks New Ground on Tree Liability. Apparently in Virginia a tree owner can now be held liable for damage caused by their tree and can forced to cut back roots and limbs if the tree poses a risk of actual harm or an imminent danger. Fancher v. Fagella (9/14/2007):
"Accordingly, we hold that encroaching trees and plants are not nuisances merely because they cast shade, drop leaves, flowers, or fruit, or just because they happen to encroach upon adjoining property either above or below the ground. However, encroaching trees and plants may be regarded as a nuisance when they cause actual harm or pose an imminent danger of actual harm to adjoining property. If so, the owner of the tree or plant may be held responsible for harm caused to [adjoining property], and may also be required to cut back the encroaching branches or roots, assuming the encroaching vegetation constitutes a nuisance. We do not, however, alter existing . . . law that the adjoining landowner may, at his own expense, cut away the encroaching vegetation to the property line whether or not the encroaching vegetation constitutes a nuisance or is otherwise causing harm or possible harm to the adjoining property. Thus, the law of self-help remains intact . . . ."