Courtesy G. Lumis
On broad-leaved trees such as maples, oaks and elms, a bud forms during the summer at the point where each leaf joins the twig or branch. This bud remains when the leaf falls off the tree in the autumn, and the following spring when the warmth of the sun touches the bud, it bursts and sends out a new shoot at an angle from the branch. This shoot grows into a new branch, and as it grows during that summer, again new buds form along which in turn become other branches. This continues year after year until finally the tree is a fully matured maze of twigs and branches.
On the evergreen or coniferous trees, the formation of buds is different. With a pine, the buds form only at the tips of the twigs. In spruce, buds form not only at the tips, but also back on the new shoot. In the case of a cedar tree, you cannot see any buds at all.
Leaves and branches perform necessary and symbiotic functions. The branches bring water and minerals to the leaves, where food is manufactured, and then return that nourishment back to the different parts of the tree.