What should I look for when inspecting trees after winter besides broken branches?

Look for three things: broken limbs, hanging branches, and bark damage. Broken limbs are dead weight waiting to fall — cut them back to the main branch or trunk. Hanging branches are dangerous because they can swing and tear more bark or drop on someone — prune cleanly at the break or back to a joint. Bark damage from ice or snow splits is an entry point for insects and rot — trim torn edges with a sharp knife so the tree can heal cleanly, and check for branches rubbing against each other.