Answer: Bougainvillea is really a shrubby vine, so you might be disappointed in the appearance (and maintenance nightmare!) if you try to grow it into a tree shape. Some other suggestions for the containers on your patio are camellia, which can be successfully trained into a tree shape, Cornus florida (dogwood), Chionanthus (fringe tree), Crataegus (hawthorn), or one of the smaller Stewartia. All of these choices adapt readily to containers and should be happy in the summertime sunshine the location will provide.
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