It likely shared similar core things with the Northmen of Esgaroth, Dale and the alnds of the river running. Like the thatched roofs. But the Architecture of the Northmen of the East is an other topic.
So here we begin, by looking at descriptions in The Hobbit.
"He lives in an oak-wood and has a great wooden house"
It is interesting that it is mentioned to be a wood of Oak, as the Vikings main wood was oak, both used in shipmaking and building of houses. Pine wass mostly used for horisontal timbering, and oak was used in Bole-house method, with hewn logs standing up.
"wooden gate, high and broad, beyond which they could see gardens and a cluster of low wooden buildings, some thatched and made of unshaped logs; barns, stables, sheds, and a long low wooden house."
It seems that both Beorn and Lake-Town has thatched roofs. This may point towards a larger trend of Northman usage of thatched roofs.
The unshaped logs can easily be intrepeted as horizontal timber logs, which are round and unshaped. It seems weird that it would refer to standing logs, ad ive never seen them being left round when in bole-house methods.
Here we have some interesting drawings of the Houses of the Woodmen, in two variants, one fully inked and the other has horisontal lines.
The horisontal lines look a lot like log houses, here is a pic of a house that looks earily similar to this one.
Buildings like this might be what the Woodsmen used, with logs either hewn or not. Gathered in villages and homesteads like this, maybe it is something like this that the Wolves in the hobbit wanted to attack?
"Soon they reached a courtyard, three walls of which were formed by the wooden house and its two long wings."
I wonder how the hall was really shaped?
"Quickly they got out boards and trestles from the side walls and set them up near the fire. ..."
Note text below.
"which the dogs took and quickly laid on the trestle tables. These were very low, low enough even for Bilbo to sit at comfortably. Beside them a pony pushed two low-seated benches with wide rush-bottoms and little short thick legs... while at the far end he put Beorn's big black chair of the same sort.... These were all the chairs he had in his hall, and he probably had them low like the tables for the convenience of the wonderful animals that waited on him. ... The other ponies came in rolling round drum-shaped sections of logs, smoothed and polished... "
Trestle tables, of the sort of the pictures below have been used a long time across europe.
Te rush-bottomed benches are interesting, we see a similar rush-bottom weaveing on Bilbos chair in the shire. Beorns black chair also has a rush seat. Both the bench and Chair appear to have short thick legs, and the chair likely has posts of a similar kind. The blackness of it is interesting, maybe it is out of a black wood or painted with Tar?
The low log seats talk for themselves.
We have two depictions of the Hall of Beorn, and they vary a little bit.
The pillars in this version have side beams, and thicker parts around their foundation. The submersion in the floor isnarrower then in the later pic. The hall itself appear to be made from standing logs, qute thin and hewn on the inside. This was a popular method for the vikings, and used where Oak was available. Either that or they are inner panels of a timber wall. This could be likely, as it would insulate from cold, and look good.
The fireplace is similar to the other one, with a guard maybe made from iron? the posts look similar to the great metal cleat at laketown, maybe they are both dwarf-made? But it says that few things save the knives were of metal at all, so maybe they are made from stone? or a hardy type of wood, although that seems unlikely.
This is the later, more canonical drawing of Beorns hall. The first thing to note is that the walls are smoth, maybe due to Beorn having whitewashed them? It was popular in the halls of the Vilings, and would serve both a decorative and a functional (it is mildly antibacterial.).
The submersion is now wider, having swallowed the fundations of the pillars.
There is also a further submersion, where the firepit lay. it is lined with stones.
The table is like the one on the other pic, with X-shaped trestles who are detachable, which was the fashion of swedish peasants for well into the 18th. century.
The low drum-shaped seats are also there. But not the rush-bottomed benches and Beorns seat of the same sort.
The back also shows us a very interesting thing, what looks like a walled room, which looks like it has horisontal logs, very thin ones. It may be the private lodging of Beorn, where he keeps his bed? Or mayby a storeroom?