I live in Boulder and we have a fence shared with a neighbor. It is falling over and being pushed into our property by a tree on the neighbors side.

It depends upon who built the fence originally, that is the owner of the fence. Otherwise its up to an agreement between owners to take care of it. If its built entirely on their property you have nothing to do with it, its up to them. You can build a fence barely on your side and leave them to clean up their old fence. If the fence is on the border between, you can tear it out, replace it, and sue them for contribution. Call me if you need to pursue this. Call a surveyor to determine the property line, do not consult county records or maps for that, even arial maps are not to determine boundaries accurately.