Should I add anything to the water in my Christmas tree stand?

Just use fresh lukewarm water. It's not clear that additives help much.
 

No; While this works with flowers, it doesn't have much affect on trees. Cutting on the diagonal would also make it hard to anchor your tree in the stand.
 

A freshly cut Christmas tree can drink as much as a quart of water a day per inch of trunk diameter.
 

Most of our growers cultivate their trees using stump culture, in which the bottom two rows of branches are left on the stump. This allows a new tree to sprout up from the stump, so the tree isn't killed when it's cut. All of our Christmas tree farms benefit the environment by stabilizing the soil, creating wildlife habitat, absorbing greenhouse gases, and emitting oxygen. Christmas trees are also recyclable.