I hope an actual Inuit speaker chimes in about the actual prevalence of trees in the arctic, as I suspect the question might be moot when someone states that they have trees here and there right up to the arctic ocean, but in the meantime here's an answer. I have a few dictionaries of Inuit languages here, and I downloaded another one and here's the results from a quick search through it. It's a western dialect of Inuvialuktun, the book portion is titled Inuinnaqtun to English, created by Gwen Ohohak, Margo Kadlun, and Betty Harnum, adapted from Ron Lowe and others' dictionary Kangiryuarmiut Uqauhingita Numiktittitdjutingit (the one I have on the shelf).

Qijuvik - Spruce Tree

Amiraq - bark of tree

Napaaqtuq - stands upright; is upright; tree

uqpik - willow tree

avaalaqiaq - dwarf birch

This shouldn't be surprising because there actually are trees in the arctic in sheltered areas such as south facing slopes or along the banks of large rivers even out almost to the arctic ocean, and Inuit communities actually range a good distance south well into the subarctic. In other words, while some communities live in places where they don't see trees, only wood, there are many exceptions that live in the sub-arctic rather than only the high arctic.

That said, in my experience, any regularly sourced important item for a community has a body of knowledge around it, even if it is secured from a distance beyond a community's normal range of travel, and even for those communities of the high arctic without trees, it's hard to imagine that knowledge about a very important resource that was known to neighbouring related communities to the south would not also be carried and shared with people further north.

edit: Just to add, it's not like this is driftwood that comes from just anywhere, in the Western Arctic, my understanding is the vast majority of it comes down the Mackenzie River, with other rivers playing a similar role in Hudson's bay and so on. People in the north are extremely knowledgeable about currents and shifting ice and such, and driftwood's availability is just a part of this knowledge. At Tuktoyaqtak there's enough of it that people have wood stoves for heating.