The White Mulberry's twig at the right shows an oval bud in the axil formed where a leaf's green petiole (it's "stem") connects with the gray-brown twig. Each tree species has its own unique kind of bud.
Notice the light-colored bumps scattered all along the twig. They are lenticels, which allow interchange of gases between the twig's internal tissue and the atmosphere. They're like tiny windows helping the tree to breathe! You can see some smaller lenticels on the Sycamore twig below and to the left. Different twigs also have different kinds of lenticels.
In the picture at the right, can you see that thin, horizontal ridge at the base of the scar, extending about two-thirds of the distance across the twig's diameter? That's a stipular scar and some tree species have them, and some don't. Stipular scars are important in twig identification. Sycamores and members of the Magnolia Family have stipular rings -- scars completely encircling the stem. The picture at the left shows stipular rings on a Sycamore twig, as well as the brown, shriveling stipule, right before it drops off, leaving a stipular ring at its base.
At the right you can see a Yellow Poplar's large stipules. These will all fall off, leaving scars. Stipules protect leaves and stems when they are very small, not yet unfolded. Once the leaf or stem is mature, the stipules are no longer needed, so they fall off.
Some twigs have weird adaptations you'd never expect them to have. For example, below, on the twig of a Sweetgum tree from next to my trailer, you can see that sometimes twigs bear brown "corky ridges" or "wings." No one is sure why some twigs have wings such as these, but several species do. Maybe they just make it harder for leaf-chewing caterpillars to travel from leaf to leaf, or maybe they help twigs dissipate heat, or hold heat.
Anyway, they're there, at least on older twigs, and such features help you identify trees even without flowers, fruit and leaves. In the local library you should be able to find several goods books showing the secrets of twig identification. Look under "winter botany," and if that doesn't help you, simply go look at the shelves holding tree-identification books, and find one illustrating buds and leaf scars.