Nope. Just a little harder to clean up when there is snow. Trees adapt to changing temperatures and actually keep themselves warm by transferring water out of cells. When the water between cells freezes, it releases heat. If the water in the cells freezes, it'll kill the tree. So trees can freeze, but it would have to go from warm like right now, to sustained below zero temps, but as long as there are no cracks or large exposed spots where water can get in, they'll be fine. When it snows, look around the base of a tree and you'll notice that the snow has melted more quickly around it.
Also, woodchips make a great insulator. Always should put chips/mulch around trees especially young ones to help keep them warm. We've moved piles of chips in the winter, and in the middle they are warm. Never tested it, but you also could probably keep some snow through the summer, if you buried it under enough chips.
Use sharp chains and a properly powered saw and you'll never know the difference.
