SW,
Some trees have pretty well behaved root systems (River Birch or Betula Nigra comes to mind) and are considered good "foundation" trees (trees you can safely plant near a foundation) or street trees. Others are a nightmare because of invasive root systems or lots of surface roots (many maples come to mind here). In other words, the answer to your question really depends on the type of tree you're talking about. Also, some trees are much messier than others and would cause you endless headaches regardless of whether they would potentially harm the pool structure

A good local nursery should be able to give you good advice on this.