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Good afternoon, thank you for contacting Just Answer my name is***** am a solicitor and I will be assisting you today. I am sorry to hear about your circumstances, with regard to the tree, this is a very common question.
A person cannot make an adjoining property owner cut the trees per se but please see my later comments about trespass and nuisance.Regarding trees, a complaint can be made to the council and they will deal with them under the nuisance tree legislation but only with regard to the height. They want £300 fee to start the process and if they find that the trees are a nuisance, they can compel a tree owner to cut them down to 2 m high.With regard to branches and roots growing boundary, these are nuisance and trespass. The overhanging pieces can be chopped off as can roots growing underground but they do not belong to anyone other than the tree owner, so the pieces should be given back although if you do not warn your neighbour first then this would not be good for already fraught neighbour relations.
If you are not capable of doing this yourself or do not want to do it yourself, you can charge the neighbour for the cost of doing the work provided you give the neighbour noticed that if they don’t trim it, you will have to get someone in to do it.You are not allowed to kill the tree or cut the routes back so far that you destabilise it.The following are link will give you some reading with regard nuisance trees.http://www.gardenlaw.co.uk/trees.html.
Hopefully I have answered your question but if anything remians unclear please do not hesitate to contact me.