Yes. But it assuredly belongs to someone or something. So you'll need to ask permission to take pieces.

The tree does grow on someone's property but it is officially city property.

What does that mean? Where does the tree arise from...and what do you mean by city property? It it planted on common land but overhangs into a private property?

In NYC the city plants trees on private property. If you want to prune or cut the tree down you need to get a city permit for it. Damaging the tree without a permit can cost the homeowner and the tree service company a few thousand in fines. If the tree is damaged or dead it is also the obligation of the city to remove the tree. Sidewalks or driveways damaged by the tree is also the obligation of the city to repair.