In AK where our 4WD club had runs every weekend - we would get our Jeeps and Jeep-type vehicles into snow up to three feet deep. Generally speaking - the narrower, lugged tire( R-1 type ) moved the vehicle better thru the deeper snows. It dug down thru the deeper snows because of its narrow width and the lugs in contact with the frozen ground provided the necessary traction to move the vehicle forward.
Wider tires whether lugged or not, floated and could not get the necessary traction when in contact with only snow.
However - some times the really wide - Terra tires - would float, get sufficient traction when only in contact with the snow and move the vehicle forward if the vehicle was light enough. This, however, could be a significant problem - usually this type of tire/vehicle would get seriously stuck - sooner or later. The tire would encounter deep, soft snows - the tire would dig thru the deep snow - contact the ground and not have sufficient traction to push thru the snow. The immense width of the tires requiring more force to push thru the deep snow than contact with the frozen ground could provide.
Whatever tire is chosen - contact with only snow is never going to provide the motive force that contact with the ground can provide.
Then there is the situation where snows are so deep that high centering will prevent any type of tire from contacting the ground - generally speaking - you be stuck.
Then there were those Jeeps with the speciality tri-angular track systems on each wheel. They too had to be very careful - they could get to areas where - when stuck - no other 4WD vehicle of any type or tire configuration could get near enough to provide any assistance with their stuck condition. They - generally speaking - were the "cool dudes" and leaders of the pack type. It would really piss everybody off when they would forge ahead of the group - a mile or so - get stuck and then expect everybody to bust their hump and come up and pull the fool out.
In conditions where the "word of the day" was - for God's sake keep it moving forward at any costs - - you dam well better be looking for less snow depth because you could be moving straight into jaws of SERIOUS STUCK.
Looking back on all those years of "play in the snow" - - I'm truly amazed that we didn't have more serious heart attack situations - digging vehicles out of grievous stuck snow conditions.