It took me 12 years to realize that with the main tree in Whiterun - Gildergreen, it's not all that simple. And you can not just heal it but completely replace it with a new tree. To accomplish this, approach the priestess of Kynareth and accept the quest to heal the tree. In this quest, you'll need to confront a Hag raven and retrieve a blade called Nettlebane, which will be essential later on. After that, return to the priestess. Following a brief conversation, she will direct you to a cave to obtain sap from the Eldergleam Tree, which will aid in healing the Gildergreen. After this conversation, a pilgrim should approach you; be sure to bring him along. Upon entering the cave, use Nettlebane to clear the roots blocking the path. Once completed, resist the temptation to collect the tree sap. Instead, approach the accompanying NPC.