Is this tree potted or in the ground? Either way, they will usually always "bring friends" (often aphids and sometimes scale or mealy bugs). The DE would theoretically work but it will take time and IMHO completely dusting the tree with it (beyond just the soil) has a risk that the stuff gets airborne and you end up breathing it in (which you definitely don't want to do).

If potted and depending on the size of the pot, you might be able to just submerge the pot as far up to the rim as you can, in a large muck bucket, and let it sit for awhile to force them out, and then pull the plant out and allow the pot media to drain fully.

In the past I have also used a solution of sugar-water/borax (you can search for the recipe on the web) and baited with cotton balls near areas of activity (make sure that you don't have it baited so much that excessive boron is leached into the soil). The scout ants would eventually take the concoction back to the queen and that will basically begin to kill the colony. The process might also take a few weeks but it also helps to ensure that the queen is dead. The most drastic method might also be to repot and be done with it!

Again...depends on whether you are talking about a container plant (and how big) or in the ground.

Well drowning seems like a good idea but ants are generally well adapted to water invading their space. They do not have lungs but have pores on their body to take in air and they can close that and ride it out. You may have to wait quite a bit - how long I do not know. Here is a fun article on how some ants deal with water: http://yourwildlife.org/2014/02/how-to-beat-flooding-ant-style/

Once I had this cool idea, I thought, to dissuade ants from climbing up the trunk. So I put electrical tape around the trunk several times with the sticky side out. The next day the whole tape was covered with dead ants and the rest of the ants were walking right over their dead brothers - problem solved.

I recommend that you pick up a package of TERRO liquid ant bait stations. They are spectacularly effective as long as your ants are sweet eaters. Read the directions on a tray, then place it on the top of the potting mix. Then leave it alone.

Ideally, the ants will swarm the tray, taking the liquid down to the residents in the pot including the queen. The ants will die after eating the bait, but you must not disturb the ants or the tray until all activity ceases.

Terro liquid ant bait trays.