Liability of the Government

I believe the government is liable for injuries people sustain that result from government negligence. I am not sure if the government could enact a law holding itself harmless. Can it?

In New York City, I believe there is a law that holds the government immune from lawsuits for slips-and-falls unless the City had previously been advise of the sidewalk defect. I believe that there is also a group of lawyers that annually advises the City of all the sidewalk defects.

The City is also sued by people hit by falling branches of trees on City owned land. Can the City or the State enact at law to hold itself harmless from such suits (sovereign immunity?)? Can the City or the State require that it be put on notice of the offending branch in order for it to be liable -- similar to the sidewalk example.