Can you plant a tree in the same place where you dug out a dead tree?

Yes, however you need to be sure the soil is ready for a new tree. The stump from the previous tree should be removed, as well as the major old roots. Once you get the bulk of the stump out of the ground - the major tap-roots will become more obvious.. and you can pull those out. After that is all removed, line the bottom of the hole with compost and root treatment before planting the new sapling.