If the tree truly straddles the property line, then you and the neighbor have joint authority over it, and share responsibility for it. A survey can determine exactly where the property line is in relation to the tree.
If the tree or any of its branches pose a hazard, then the tree, or the hazardous branches, should be removed. If the hazardous branches overhang your property, then you are within your rights to remove them. If you feel that any part of the tree that is on your neighbor's property needs to be removed as well, and your neighbor will not agree, then you may need a court order to do so.
You ask if your neighbor can take action against you if you decide to remove the tree. The answer is yes, if the part of the tree you remove is on his property. Your neighbor may seek a preliminary injunction from a court ordering you not to remove the tree, or sue you for trespass after the fact. Whether your neighbor will be successful is unclear, but he/she certainly could bring a lawsuit.
Keep in mind that this is not legal advice, and I highly recommend that you seek the advice of an attorney in your area before taking any action.
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Law Offices of Marshall D. Chriswell
714 Philadelphia St., Suite 200
Indiana, PA 15701
Marshall D. Chriswell is a civil practitioner with offices in Indiana and Clearfield Counties. Mr. Chriswell's practice emphasizes Wills & Estate Planning, Probate, Real Estate, and general civil disputes.
Mr. Chriswell provides free initial consultations, and will gladly meet clients in their homes upon request, including on the evenings or weekends.
This communication does not constitute legal advice and does not establish an attorney/client relationship. If you are seeking legal advice, feel free to contact our offices.