Always wondered about this. Been looking for some Ohio collecting areas. My conclusion was that public land ODNR does not permit collecting plant material and private land we need permission.
I would check their DNR website. Another idea would be to find out what plants they consider invasive. No one will be mad if you are pulling invasives right???
OK, first off, this link has the phone numbers you need.
https://www.uaf.edu/files/olli/08transplantingtreesandseedlings.pdf
This can also get a little more confusing because of how the contracts between the state and various tribal entities are framed up. When in doubt, don't, as the saying goes. Private land should be ok, but be sure you can prove provenance, whether you are leaving with it by plane or boat, or shipping it. Law enforcement of tourists taking natural resources is spotty at best, but if you get singled out, make sure there are no doubts else someone WILL try to make an example of you.
Driving with them through Canada is a solid "no."
Now if this were an animal product: Every hunter and fisherman in AK is a self-styled deputy of Fish&Game, because F&G are kinda feared up there and nobody wants to get caught out as you are assumed guilty until proven guilty. Plant products have come under more scrutiny in recent years because of poachers harvesting things en masse from Native lands.
As for what you can find in Alaska varies hugely by area, with more variety in South East, tapering through south central and dropping significantly as you get north of Fairbanks. Swamp Spruce is kinda the filler tree everywhere. Clumps pop up if you forget to mow.
All that said. This is early winter in Alaska. In the Anchorage area, the ground is already frozen by Halloween and there is usually a foot of snow on the ground.