Question
What is the best treatment for eliminating ants from a water oak tree located in Alabama--considerable amount of sawdust located at root of tree (circumference about 2-3 feet.

Answer
The inside part of a tree is dead and if there is an opening fungi get into the tree and over time will hollow out the tree. The living part of a tree is just under the bark and a hollow tree will survive and remain fairly healthy even if it is hollow. The ants are finding a easy place to nest and are not harming the tree. BUT if they build up they may invade the wood in nearby buildings  and this will cause problems.

Carpenter ants nest in trees in one of two situations: 1) in rotted, decayed wood or 2) in the center heartwood section of the tree. In neither case are they harmful to the tree. Control is unnecessary for the tree's health, as the ants are taking advantage of preexisting soft, weak wood to establish their colony. Insects, disease, or environmental conditions such as drought are often responsible for weakening and killing limbs or sections of trees. This allows wood rot to set in, which results in wood decay, giving carpenter ants the opportunity to colonize the tree. Carpenter ants use knots, cracks, holes, and old insect tunnels to gain access to these areas.

Control of carpenter ants in trees is warranted if there are indications that ants are entering homes from colonies in trees. If there is evidence of this, the best control is to bait the colony.
Baits, such as Terro. Baits tend to be slower-acting than other forms of carpenter ant control, but they are easy to apply and give good results, especially when the nest can't be located. The ants themselves will carry the bait back to the nest, which usually provides colony elimination.

There are a few baits available to nonprofessionals for carpenter ant control. Most retail products are liquid or granular formulations containing hydramethylnon, sulfluramid, abamectin, or boric acid. An inexpensive liquid bait of 1% boric acid in a 10% sugar water solution can be mixed at home, but it is very slow acting and must be constantly replenished. Baits vary a great deal in their effectiveness. Carpenter ants have complex food preferences, and some of the sugar-based baits will not be attractive to the ants long enough to be successful.  

If the nest is exposed  you can use a liquid or aerosol ready-to-use insecticide, such as bifenthrin, cyfluthrin, deltamethrin, or permethrin. Spray the insecticide directly into as much of the nest as possible. The more of the colony that is exposed, the better your chance of destroying it. It is necessary to anticipate a carpenter ant colony and have a product ready at the start of construction. Once the nest is exposed, that portion of the colony will try to relocate to protect themselves.