It is always best to talk to your neighbor first to let him or her know of the problem and to see if you can work together and/or share the costs of solving the problem. If your neighbor is unwilling or unable to help then you may need to take the matter in your own hands.
Your first option would be to cut back the tree branches or roots that are encroaching onto your property. Although not addressed in Utah courts, most jurisdictions allow property owners to cut back the roots and branches that encroach onto their property. Before doing so, however, it is best to give your neighbor notice of what you are planning to do, especially when needing to trespass onto adjoining property to cut the branches or roots.
Note, however, that if the tree is a heritage tree then you may need to file an application with the Utah Division of Forestry, Fire and State Lands before altering or cutting back the tree.
Your second option would be to bring an action for nuisance if your neighbor is unreasonably using the tree to cause you damage. In Utah, a lawsuit for nuisance is determined on a case by case basis and whether a nuisance is actionable depends on the reasonableness of the use of the tree in the particular locality, under the circumstances of the individual case. Cannon v. Neuberger, 268 P.2d 425, 427 (Utah 1954). As such, it may be helpful to contact a local real estate attorney to determine if a nuisance lawsuit has merit.