One very important point missed by all the answers is that the tree and all the overhanging branches and roots belong to your neighbour.
You must first ask him to cut away the branches that overhang your property and roots that traverse it.
If he does not do this in a reasonable time you are entitled to cut the branches back to the property line. After you have cut the branches you must offer them back to your neighbour but he is under no obligation to take them
This is the law of nuiscane and the principle of abatement of a nuiscane.
With regard to the tree falling and damaging your neighbours property after you have cut roots and branches, you can not be held responsible, even partly. He was fully aware the tree existed and the state of it, he was asked by you to deal with the issues it was causing on your property, he failed to do so. You then dealt with it and brought it to his attention by offering back the cuttings. If he then let the tree get into such a pecarious state that it damaged his property, that is his own fault. You are not liable.
Hope this clears thing up for you.
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