I believe the answer for New York is "no problem". Inside the USA the whole point of the interstate commerce rules is to allow unrestricted shipment between the 50 states of the USA. No state that I know of prohibits the export of trees, and the only states that require the inspection of incoming plant material and produce are states with major agriculture pest concerns. California, Arizona, and Florida are the 3 states that require Phytosanitary Certificates, and or Inspections at the state border.
If you harvested the trees on private property, with permission of the property owner, and the destination is inside of the USA, there are no issues with CITES or other Endangered Species regulations (Lacy Act).
So just put it in a box, and ship it.
If the origin or the destination is outside the 50 States of USA, then a whole bunch of rules and regulations kick in. Quarantine is mainly for living plant materials coming in from other countries. It does not apply moving around inside the USA.