What is the best way to prevent tree roots that are already in the ground from growing up through the raised garden?
Raised Vegetable Garden
You will not be able to keep tree roots from growing thru the raised bed unless you make the bed as large as the drip line and back fill to a depth of 2-3 feet, which will of course kill the tree. Hopefully you have an alternate location for your raised bed for vegetables in full sun.
Depending on the sisze of the bed, you may be able to line the bottom with concrete or brick to a depth of 4", and this might prevent some roots from growing through.