Just a note, but the vast majority of the Earth's photosynthesis isn't from trees, but from plankton in the ocean. Most photosynthesis would not be stopped by nukes dropping in the ocean because no one is going to target the ocean. The changes to the ecological system would have unknown effects, though, but it is certainly within the realm of possibility that any changes didn't completely undermine the necessary photosynthesis for the air to remain breathable.
Just because we see dead-looking trunks above ground doesn't mean the trees are actually dead. It is very likely that some of them are still alive just as trees today are alive during the depths of winter.
One type of tree that is specifically named in FO4 is maple trees. There are many varieties, of course, but in general maples are very hardy trees and it is not beyond belief that some would survive in the wasteland.