So, I got this small potted olive tree for Christmas. Soon after it lost all its leaves. I assumed it had died, but a couple months ago, some leaves appeared again. Since, I began to water it again (not too much, since I assume I shouldn’t), I put it outside, and replanted in in a bigger pot. However, the handful of new leaves died out, and since no new ones appeared. The baby tree seems completely dead.
However, I would want to be really sure it is before giving up and throwing it away. How can I know?
Try bending a small branch. If it snaps off, that part is dead. If it bends or doesn’t break cleanly, it’s still alive.
What pudytat72 said. Start testing branches from the outside in. All branches might be dead, but if the trunk itself is still green/white inside insterad of brown, it might still be alive.
If you don’t want to break off branches, try repotting the plant: take it out of its container, shake loose the rootlets and put it in a larger container with new soil. If all roots and rootlets come off easily, or if they smell bad, it’s because they have rotten away and are dead. Dead roots, dead plants.
If you want to make sure it’s dead, run it through a wood chipper. 
Then you’ll know.
Scratch the bark on the trunk with your fingernail - not much! and if it is green underneath, it is dormant, not dead.