As far as I know, the only Eucalyptus tree's are from Oz. But they're quite popular in the U.S., at least in Calif.
That's very close ts, except it was a huge 2-story house not a cabin. And aside from the house, a clearing between the house and shed, and a driveway, everything else was the woods. The two eucalyptus were the only non-native trees on the prop. P.s. there was also a gold mine on the prop, with a cabin built on top of it. The gold had been mined out and the tunnel entrance re-filled with dirt, as mandated by law. Visiting that old cabin several times reminded me of how easy we have it to-day. The cabin had no elect. or gas or running water. The nearest water was several hundred yards away and had to be brought to the house by hand. And if you had to crap or pee you had to go outside, even in zero temps during a blizzard. Light was provided by candles or lantern, heat by a wood-burning stove (at least wood was plentiful) and all supplies had to be carried by horses or people. The nearest place to get supplies was 20 miles away, again by riding a horse, horse drawn cart or walking.