Thank you for the information and your question. Normally, under Colorado law, the neighbor who is the encroached upon neighbor (the one not owning the tree) has the right and responsibility of self-help. That means that they can trim any overhanging branches from a neighbor's tree as long as doing so does not kill the tree or make it unstable. If they do not take those steps, then they are generally precluded from bringing suit for any damages that could have been avoided had they exercised their right to self-help.
On the other hand, if the tree in question is diseased, dying or unstable and the owner knew, or should have known that, then they have liability for damages to their neighbors property as a result of negligence in not trimming dead branches or cutting the tree down, if necessary.
So, the key in your case is whether the branch that fell was dead or diseased and you could have had notice. If so, then ultimately a court would order payment to the neighbor. You can see a general discussion of CO tree law issues by going to: http://www.hindmansanchez.com/resources/article/what-do-when-neighbor%E2%80%99s-tree-strays
You might want to hire an arborist to look at your tree and perhaps call your homeowner
's insurance company about this claim.
Please let me know if you need any clarification. I would be glad to assist you further if I can.