I don't think you need warm air for that. I know the effect, and it's quite common in the Netherlands, we even have a word for it: "ijzel." (cf. "ijs", ice)
It is possible for rain to freeze coming into contact with a cold surface, but in general, trees and such don't store that much heat (or cold) in them sufficient to let the rain freeze.
Instead, the rain is typically undercooled for lack of any nucleus for crystallisation. The rain is of freezing temperature, but doesn't know where to start. As soon as it touches anything solid, it will freeze instantly and cover everything with a thin layer of ice.
It's one of the worst things you can get on the road in a moderate climate...
Henk
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