It's not advisable to grow a tree that drops things next to a pool. The hackberry and pistach may drop fruit. The autumn blaze maple will produce samaras (helicopters) to disperse their seeds. Although it's an ornamental tree rather than a shade tree, the fruitless spring snow crabapple is a clean tree.

Growing a tree near or on your poolside should really be lovely. However, this might cause you more efforts in cleaning the pool every time especially that trees regularly shed leaves off. Good if we only have leaves that are solid as dirt in the pool. But we all know that there are some substances from trees that may color or dissolve in the water causing you to change the water which surely would not be convenient. Why not just use sun shelter or sun sail awnings to add shade.