People are seeing rodent damage on trees, especially young fruit trees. The worst-case scenario is if the rodent (whether mouse or rabbit) ate into the heart of the center wood circling the entire trunk, there is not much hope for survival. If the damage is less than ½ of the trunk, there is a good chance that it will be OK. If more than ½ way around the trunk, you might have to prune back the branches to reduce stress on the tree.
Prevention for young trees would be to use plastic trunk protectors when you plant the tree. This prevents rodent damage and sunscald. Once the tree is a few inches in diameter, the problem from animals is usually not a threat.
With this last snow, there is also some branch breakage due to the weight of the snow. If the branch is broken, cut it back to the nearest branch crotch or bud. If you need to remove the whole branch, cut it back to the trunk leaving the branch collar (the swollen area on the trunk) to heal.
