Both the Faroes and Iceland had trees before the vikings arrived and deforested the land, with birch being the most predominant species. But yeah, the wind makes it hard to bring a forest back once it's gone. The rewilding youtube channel Mossy Earth just released a video a couple days ago on their efforts to bring back some birch forests in Iceland:
Was it just for fuel? I don't imagine birch being sufficiently large for shipbuilding. Were there larger diameter trees?
Fuel and settlement activity (shelter, fencing, farmland, etc), I don't think there would have been much wood worthy of shipbuilding, at least compared to their traditional ships.