The Green Keeper is fought three times. Once in the Cave to Yggrasil and twice inside Yggdrasil (in the Underground level and in the Upper level). Natural Healing will heal some HP at the start of its turn. Its attacks are the same in all encounters, but its main trump is its Natural Healing.
The Green Keeper can be defeated with continuous physical attacks. However, magic is recommended as it's quicker and easier to finish him off, though he will counter it with Osmose. At some point it will cast Haste on itself. Once he is below half HP, do not cast Slow on him at this point, as he will respond automatically with Haste. The first and second Green Keeper should not be much of a problem. However, the third one has very high Natural Healing. The combination of Haste and more than 400 HP of Natural Healing can mean disaster. If the player cannot beat him, put more Mages on the team to spam Fira or do some Level grinding.