My name is ***** ***** I am a NSW Solicitor. Thank you for your question, and will do my best to assist you with your question. Please understand I am very limited on what I can tell you and that you may have to get more detailed advice.
She has the right to cut i down but she can ask you to remove it. If it falls on its own you are liable for the damage that it causes. She can also seek you pay the costs of removing it.
If she decides to take you to court, the court can and possibly will order you to cut the branch down. It may also require that council approval be granted to prune it as many councils have tree preservation orders.
Any damage that it causes to anything will be your responsibility.
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