There are trees owned by the City but it will usually fall to the property owner to maintain it and the liability will probably go to the property owner also.
Sidewalks are covered under a "prescriptive easement". The responsibility to maintain the sidewalk from trip and slip hazards is court tested and common.
As suggested call 311 and ask what you are expected to do but don't be surprised if you get the buck passed to you. If you feel that the info given to you by the City is wrong, then contact an attorney.