I have pools at two houses and two condos, I do not suggest any trees other than something like a green giant or another Arbivida. You do not want to be getting leaves out of your pool all the time, do you? I love trees, I love the shade, but I do not like them near the pool. Yes, they have machines to pick them up for you. Yes, you can vacuum them. Yes, I can be lazy. However, if you want shade, use a shade sail or something like it.

https://www.amazon.com/Shade-Sails/b?ie=UTF8&node=3480728011

Before you get a tree, think of acorns in your pool after a heavy wind or rain. Think of blossoms getting in there during the spring. The flowering trees can be a nuisance --- bradford pear-crape myrtle-etc.

I know, the water gets warm during the summer. That is why I suggested the shade sail. I have seen it used at some fabulous pools around the country.

However, all of this is my 0.01.