Can anyone advise me about tree trimming?

We have a big old oak tree in our side yard. NYC will not allow us to remove it. Despite that branches fell in Sandy. We trimmed it but, we feel it must be trimmed again. We want it trimmed because we feel it is a hazard and because we recently had solar installed. We had a few people look at it and they all approach it differently. They all say that if, we remove too much it could fall. No one is clearly telling is what to remove. They are all recommended on Angie’s list.