Note that I am assuming a few points that weren't made directly in your message, most significantly is that you imply that this is not a 'border' tree with shared ownership.
In Canada, there have been a variety of cases where a tree owner has been held responsible by the courts for harm and costs involving unreasonable nuisance. There are also cases involving the right to 'self-help'.
Again, assuming a few points, I suspect you are trying to imply that the removal of your tree should be done with costs shared by your neighbour. I can't imagine why this would ever be deemed in part his financial responsibility. It's your tree!
While he has had the right, without causing undue damage, to prune the overhanging branches of the tree back to the property line during its life, this does not obligate him to do so.
As said, in Canada there are a number of examples of cases where the legal principles involved are explained. An excellent book sharing light on the subject is Arboriculture & The Law in Canada. In the U.S., I believe a similar version was published entitled with the shorter, Arboriculture & The Law.
Of course, the most accurate source of 'truth' for your question will be to ask a qualified lawyer in your jurisdiction. However, based on reasonability, I anticipate that your neighbour would have no financial obligation to help you pay for the removal of 'your tree'!
All the best!
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