If trees did not shed their leaves, their soft vegetation would certainly freeze during winter time, damaging and, no doubt, killing the tree.
This is just plain wrong, and makes no sense. If the leaves froze and died during the winter, how is this different from the tree losing the leaves to begin with?
Trees lose their leaves to stop the process of drawing water up from the roots through their trunks. If water were to go through the freeze/thaw cycle inside the long "straws" of the tree trunk, the tree would be torn apart from the expansion/contraction cycle of the ice.