I went to the Parks Department in Flushing Meadows Corona Park where the truck full of confiscated bicycles sat to get the scoop. Here, Captain Kenneth Brown informed me that today began a citywide campaign involving the City Parks and Recreation Department. This time all bikes chained to city trees or to the metal fencing around those trees will be seized in neighborhoods throughout the 5 boroughs. Bikes are then taken by truck to the nearest Parks Department Headquarters where they are held for 2-3 days before being sent to a depot in Long Island. In order to retrieve your bicycle, you must pay a $50 summons - that is if you can find it.
According to the Captain Brown, bicycles are only considered to be parked "legally" to bicycle racks.
I agree with one of the commenters that locking the bike to the tree can be harmful to the tree, but locking the bike to a fence, shouldn't. And the city really should provide more bike racks.