Some healing goo hit the lich and then he became this fleshy pink thing that I suppose looked like a giant child and the Finn and Jake took him back to ooh and tree trunks and mr pig adopted him as their son and called him sweet pea I think, you kinda gotta watch the episode
The lich king got turned into a big softie by repressing his evil. He needs tree trunks kindness to keep the lich at bay. So they adopted him. There are a few episodes where sweet pea gets upset and turns partial lich. But watching the episodes is the best way to understand.