I remember studying about how to determine a tree's height, given the sun's angle and the length of shadow it casts. I did a quick google and you should be able to figure it out with a little algebra and ratio and proportion.

"One of the simplest ways to measure a tree's height requires a sunny day. Pound a stick or dowel into the ground. Measure the length of the stick above the ground, then measure its shadow. (For example, the stick might be 3 feet tall; its shadow might be 2 feet long.) Now you know that a 3-foot stick casts a 2-foot shadow. Measure the shadow cast by the tree. If the tree's shadow is 10 feet long, you can use a little algebra to figure out the height of the tree: If 2/3 = 10/x, then 2x =10 X 3 or 2x = 30, so x = 15; the tree is approximately 15 feet high."

Of course the angle of the sun above the horizon will vary depending on season, and your latitude. Pick a sunny day, do it at the hours you do not want the shadow to hit your pool area. Use a yardstick and find out the ratio of shadow to a known height at those times. If you want to plant a tree about 22 feet in height, and the ratio of shadow to yardstick height is 2:3 then your tree would throw a 15 foot shadow. This shall be different at different times in the year, of course since the sun is most overhead not only mid-day, but in summer. So, plan ahead and do it in fall or spring X number of weeks away from the equinox. It should be about the same deviating one way or the other. It's very ballparky and yes there are charts with exact information in them, but it can give you a rough idea.