It may need more light. Those plants love full sun and depending on the amount of light under those trees it may not be getting enough to support good bloom production.
In the tropics bougainvilleas bloom almost continuously. There the day and night cycles are more even. But in some parts of the gulf coast region our long summer days seem to discourage them from forming the blooms we so want to see. They bloom better for us in spring, fall and even winter if they are in a warm greenhouse. Their bloom cycle lasts a little over a month. After this the blooms or bracts fall off and the plant cycles through a period of a 6-8 weeks during which it grows leaves and shoots but not blooms. Then it will often enter another bloom period.
Bougainvilleas bloom best when they are not in too vigorous of a growing condition. Potbound container plants and in ground plants living in marginally poor conditions seem to be among the best blooming specimens. Cut back on fertilizer and water enough to keep it going but not luxuriously growing. Give the plants only enough water each morning to prevent them from completely drying out. Don't worry if the leaves drop your bougainvilleas will still be okay. Continue this procedure for two weeks, then begin watering daily, and fertilize lightly with a product with a 3 1 2 ration such as 17 7 10 or 15 5 10. The plant should settle down and start blooming in time.
Some people take the drastic measure of root pruning to get their plants to bloom. This is done by forcing a shovel into the soil in 4 locations a foot or two away from the base of the plant, thereby cutting some of the roots. I would only do this as a last resort, such as if it wasn't back in bloom by next spring.
In Florida bougainvilleas should be pruned after the winter bloom, usually in February. Pruning consists of cutting back the lateral shoots of the previous summer's growth to within two buds of the base of the past year's growth. Without this pruning, there will be few blooms.
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