I'm not an arborist or tree expert and I tend to be a bit paranoid about trees close to pool installations, so my thoughts are not be taken as professional advice. Trees provide shade and that may be a good thing if pools in your area tend to get too warm in the summer. Trees also provide leaves, nuts (acorns) and miscellaneous debris; spiders and other insects; the animals that eat insects or nuts or leaves, including birds, squirrels, rats, [o]possums, snakes; and their excretions. If the trees are within 10 or 15 feet of the pool, these items ARE going to be distributed into the pool and that means an increase in maintenance (leaves/bark stain pools) and higher chlorine levels, to beat back (oxidize) the additional organic load on the pool. To determine if the pool will be directly threatened from roots I suggest determining the typical root ball size for the tree species closest to the pool and estimate its growth over the next 20 years. If during the dig you notice roots being dug out at the perimeter of the hole, remove the tree or plant that put them there.
I have a favorite sequoia sempervirens that I planted 15 yrs ago and an even older all-in-one almond tree, both less than fifteen feet from the pool at the time of build, that I refused to remove. But after only 3 years, I see the folly of this decision and will probably have to take them out in the next couple of years. Just my 2 cents.
Thank you.