Do backup generators kick in when a circuit breaker trips

Obviously, they would kick in during a total blackout, but what about during a 'partial' blackout? i.e. When a circuit breaker trips and power is cut in a specific area of a building.

And the Bonus Question (tee-hee)... Typically, how branches would a circuit breaker have inside a building such as a school? A branch for every room? Every floor?

Thanks :)

There is no single, one size fits all answer.
Generally, a generator would start automaticly unless there was some reason not to

Generally, generators serve an entire building or section of a power grid. The entire building or grid goes down at once and the generator picks up the entire load

Examples of exceptions:

A pumping station for liquids (water) would not start automatically. Pressure surges resulting from the pumps losing power even for a second and then getting power back and restarting would break the pipes. So, when the power goes down, the operators of the pumping station shut the switches to the pumps, close the valves, then start the generators. The pumps are turned on one at a time and the valves opened slowly

Multibreakers protect circuits, usually a section of a building. Low voltage things like lights, there are several on the same circuit. High voltage/current things like stoves and air conditioners have their own multibreaker and circuit