You need a third party filter, I think. i don't remember the stock particle systems having alpha interactivity. DigiEffects was once the goto provider (but I'm sure there are more modern offerings from the likes of Red Giant.
CC Particle World has a floor (which can be positioned to be a wall or ceiling) that can be turned "sticky" but it is a flat plane. You can access those controls with the Options button. Very confusing first few times you go there.
particle play ground will let you use a layer map to effect particles... and one way would be to use the layer map to effect particle speed and have them 'stick' to the layer map. you would need to create a layer map that looks like the parts of the trees that snow would stick on (branches and such).
particular (from trapcode) is a particle system that will allow you to define a layer as a wall or floor and use the layer's alpha to effect particles. you'd set the physics model to 'bounce' then set the plane (either floor or wall) to the layer that you want to define the areas where snow would stick, and set the collision event to 'stick'. then particles that hit those areas will 'stick'.
[david bogie]"Particle Playground, the ugly little brother no one wants to choose for basketball."
haha... yep, i think particle playground suffers from being overly complex... you can have 4 different particle systems (cannon, grid, layer exploder and particle exploder) all going at once from just one effect and they can effect each other... plus, the best way to drive its animations is thru layer maps, which aren't the easiest things to get used to, and you can use around 4 different maps, each with 4 channels... they probably should have made a simple version to ease the learning curve, maybe particle swing set or particle monkey bars before letting you play on the whole playground.
once the cycore effects were added to the pro bundle, i found i've rarely needed particle playground, but for snow to sticking trees, it's the only one i could think of that might be able to give that type of control.