Trees use a line of cells, called abscission cells, to essentially cut the leaves off from the branch. Leaves die if left on a deciduous tree.
The leaves of a tree provide it with nutrients from sunlight. In the winter on a warm day, the leaves may try to continue the process of providing nutrients, called photosynthesis. Because the process requires water, if the weather gets cold again, the leaves are frozen on the branches, causing them to die. Therefore, the trees would be stuck with dead leaves in the spring, which would lead to the tree's death. Instead, trees become dormant in the winter by pushing off their leaves. They grow a new set in the spring. In tropical climates, trees shed their leaves during the dry season for similar reasons; they would die on the tree. Only deciduous trees shed their leaves; evergreens keep them all year round.