Tree "ownership" is determined by whose property the trunk stands completely on. If the tree is a boundary tree (the trunk straddles a property line), it is owned jointly. If most cases, property owners may trim branches and roots which encroach on their property if the stary with certain guidelines: trim only up to the property line; do not enter the owner's property without permission; do not destroy or damage the tree by trimming roots or branches.
If you have a dispute with your neighbor, call the Mediation Program at 425-452-4019. they can help you explore your alternatives, coach you on how to negotiate with your neighbor or provide free mediation to help you and your neighbor find a workable solution.