Just because she is buying her house, it does not give her the right to dictate what goes on in other people’s gardens!
She is entitled to remove any overhanging branches in her garden, but that is about it. You could say to her that if she is moaning about the overhanging branches, or the tree itself, she is welcome to contribute to the cost of pruning it. It certainly might shut her up.
Some people who end up buying Council houses can end up having this “little Englander” attitude to those around them. They get it into their heads that because they own their house, they own the street!
Do try and take her down a peg or two, she sounds like her head is creeping too far up her arse!