In this case (PDF), the Tennessee Supreme Court described the rights of neighbors in tree disputes.
Quote Quoting Lane v Curry, 92 S.W.3d 355 (2002)
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 by it, 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 Tennessee 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.
The rule is slightly different for dead or decaying trees:
Quote Quoting Lane v Curry, 92 S.W.3d 355 (2002)
It is important to note, however, that dead or decaying trees that cause harm are in a category of their own and require a different analysis. Unlike the cases involving harm caused by live trees, which are based on nuisance or trespass principles, cases involving dead or decaying trees are typically analyzed according to negligence concepts. Thus, liability usually turns on whether the defendant landowner lived in an urban or rural area, and whether the defendant knew or should have known that the tree was dead or decaying and herefore was on notice that the tree might fall. See, e.g, Staples v. Duell, 494 S.E.2d 639 (S.C. Ct. App. 1997); Taylor v. Higley, No. 02A01-9207-CV-00194, 1993 WL 137593 (Tenn. Ct. App. May 3, 1993); see also Restatement (Second) of Torts §363(2) (1965); Dan B. Dobbs, The Law of Torts 588-89 (2000).