The natural growth pattern of a bougainvillea vine, if left completely on its own, is to essentially grow straight outward as a vine. Without encouragement, the plant will never produce branches and new stems on its own. The problem here is that the bright flower blooms of the plant only sprout from new green growth. Therefore, without a grower's helpful maintenance, the bougainvillea can potentially become nothing more than a green vine that does not produce flowers. The easiest way to produce new growth is to pinch back the old growth to encourage new, full growth.
Pinch back the stems of the bougainvillea as soon as spring hits and the temperatures begin to warm toward summer. The bougainvillea will begin to produce some new shoots. Take your fingernail or a sharp pair of scissors and remove the top 1/2 inch of soft, green growth from the ends of stems. This encourages the stems to produce two or three strong new shoots, from which flowers will bloom in the summer and early fall. You can continue this pinching exercise all throughout the summer and blooming season to encourage your bougainvillea to really branch out into a flowering tree or shrub.
Pruning, which involves removing more of the vine and generally requires larger and sharper shears, is best done at the end of the first flowering cycle, when the flowers are beginning to wilt and fade. Snip off fading flowers just above one of the nodes on the stems or branches to encourage the plant to produce fresh new growth from the cut area. If you prune early enough in the season and your summer and fall are warm and long, you may even get a second bloom cycle in a single season. At this time, you can also prune the vine for size or shape if you are controlling that as well.
During bad winters with high winds and intense colds, the bougainvillea can sometimes experience damage to stems and branches or to the vine itself. This involves emergency pruning to remove the dead or damaged areas, and this must be done before the weather gets warm and mold or insects can move into the decaying area. Prune away dead and damaged parts of the vine in early spring, once the final frost of the year has passed. This will avoid damaging the vine while keeping it from getting infected or infested.