Tateyama trekking with baby stroller

Hi, my husband and i will stay at tateyama hotel - at peak of the alpine (i believe)- with our 2yo son, in last week of this coming april. We would like to do some trekking and walking on the snow. Is it possible to push a stroller on a snowy terrain? I saw the pictures from web, even midagahara area is still covered by snow in late april. We will bring compact stroller as advised by the web, but this stroller also comes with small wheels which im not sure can be pushed on snowy road. I dont want the stroller end up stuck or sink in the pile of snow.

We come from hot weather country and has not experience with snow at all, so all advices about bring a young toddler (2 year old) are appreciated, e.g. winter clothing, suggested trek/route that is easy to visit with kids (and stroller), gears, any other preparations?

Also how thick is the snow during that period? Will it make it hard to walk if just wearing normal shoes for kid?

Thanks for your advices

I wouldn't even contemplate hiking/trekking with a stroller, let alone in the snow. Get a carrier http://www.babyslingsandcarriers.com/backpack_baby_carriers
Ideally all of you should wear snow boots or waterproof hiking boots if they're warm enough. You should do some research about snow-appropriate clothing.