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The most important thing to know is that most of the Sierra Passes will be closed at that time of year, so driving across Tioga Pass from Mammoth through Yosemite simply is not possible. You either have to go around to the south via Bakersfield or to the north via Lake Tahoe. You should also note that chain restrictions will apply in mountain areas, including Yosemite, where carrying chains will be mandatory.

Other than that, we simply can't tell you if you'll see snow or ice on the roads, because that depends on the weather, and there is no way of knowing what the exact weather will be in a given area anywhere when its still a month away.