Both trees and shrubs have a layer of actively dividing cells called cambium. This cambium is located between the bark and the hard inner wood layer. The cambium layer of both the scion wood, or the branch that's being attached, and the branch from the mother plant must be in direct contact for grafting to be successful. While there are several methods, the parts to be joined must be freshly cut with disinfected clippers or a knife. You can disinfect by soaking the blades in a solution of one part chlorine and nine parts bleach. Then join and secure the two cambium layers with grafting tape or rubber bands. The union, in some cases, must then be covered with grafting wax or petroleum jelly. If successful, the branches will knit together.
If the break just occurred and the broken edges of the limbs haven't had a change to dry out yet, it may be possible to reattach them limb. If the edges aren't too jagged, you can probably fit them neatly back together so that the cambium layers touch. Secure with plastic grafting bands. If a branch has broken away from a trunk or if the branch is large, reattach by using screws, which are typically never removed, and then use additional supports if needed to hold the limb in place while it heals.
If the broken branch is less than 1 inch in diameter, the easiest grafting method involves the whip graft. Both the scion and the limb you're attaching should be about the same diameter. Cut both branch tips at a smooth, straight angle. Then, hold the branch with the cut side toward you, place the disinfected knife horizontally across the cut surface, and slice downward about a quarter of an inch. Place the notches on each branch inside each other so that all raw parts of the branch tips are touching. Wrap graft tight with grafting tape or rubber bands. Cover with grafting wax. When branches knit, remove binding materials.
If the broken limb is larger than 1 inch in diameter, a cleft graft may be more appropriate. Cut the damaged limb back past the point where the breakage occurred. Make a clean, perpendicular cut across the branch. With a sharp, disinfected knife, split the branch end across the center, cutting down about an inch or more. Prepare the scion by making long, smooth cuts, creating a wedge- or pie-shaped pointed tip. Insert the scion into the split on the limb, carefully lining up the cambium layer of one side of the scion wood with the cambium layer in the split of the limb. Repeat, placing the second scion on the other end of the split in the stem, if desired. Cover the entire exposed area with grafting wax.