Tree stumps....

[mohawk]

Tree stumps decompose over time, it will eventually cause a sink hole, the decaying process will likely slow down once the pool is up since less moisture will get to the tree stump (no direct rain anymore), but since teh stump is organic in nature, it WILL decompose eventually.
If you are planning on having this pool for 20-30 years, I would advise you have the stump removed, however, if 10 years is a good enough time span for the pool, the stump will likely not become an issue.

If you center the pool over the stump, maybe you will get a deep end over time!

This is exactly what the pool installer told us. We decided to move the pool 3to 4 feet away from the tree site.

[robot9000]

I had two area's that had stumps 10 years ago. The stumps were gone (Ground down) but there were large root structures just below the surface. When I removed about 4 in of soil, I found the roots. They were soft and springy just to stand on. I found they whent out about 3 ft in several directions. I ended up digging them up and back filling with sand and some paver gravel my neighbor had (He was building a paver patio at the time).

I used a hand tamper and watered it to get it to settle as much as possible. I had to do 2-3 cycles of tamp, water, backfill to get it all solid.