Dex, if that tree is on your property it's your responsibility and now that you are aware there is a problem and had professional advice to cut it down, if you don't you could be help liable for being negligent. ie your neighbours house insurance could pursue you to recover costs and you insurance company refuse to back you.
From what I understand you neighbour can refuse to contribute to having the tree removed unless it can be shown that the excavation for the basement has influenced the stability. Tree is on your property and if the roots extend into your neighbours property they are entitled to cut them off at the fence line.
Fences and over hanging trees are a source of many disputes but there some pretty good guidelines
