Up here in the local mountains right now there is dirt, then mud, then slush, then snow as you head up in elevation.
Stock tires, 100% useless in anything except some dirt.
I have the Michelin AC10's, very aggressive dirt/street legal tires. If the snow is more than 2-3 inches deep it's almost impossible to go uphill at all without walking the bike along. You can bounce up and down on the seat from time to time and get it to go further, but eventually you will stop. You must, at all times, decide when you are descending on a trail in snow, whether you could get back up if you have to. I found that out the hard way one time, and had to sweat bullets pushing the bike back up a slight slope all the while hoping it wouldn't overheat (since the tire is spinning while you're pushing and not much air is getting into the radiator). Just flooring it won't help either, you just dig a hole. 1 wheel drive just don't cut it...
The best solution? Honestly? If you're wanting to ride all year on most trails? A cheap used 4x4 ATV...