The name change
Because of Carter's orphanhood, he becomes eligible for a name change. His new custodians are his grandparents, and their more spiritual opinions about reality are helpful to the mourning youth. They change his name to show that his roots are always planted in love, and they seek to provide him new resources that will help him grow. Grow into what? They seek to help him become the person he was born to become. His true identity is stowed away in dramatic irony in the opening of the book, and his name change points toward the fulfillment of that identity-revelation.
The liquor
The moonshine adds a level of irony and complexity to the novel that defies clear explanation. Yes, addiction to liquor is emotionally painful and severely unhealthy, and yet, this family maintains that it is invasive and unjust for the American government to ban the chemical. This Puritanism frames spiritual devices like drugs and alcohol as obviously evil and witchy, but the Natives take that as permission to free their shame from legal obedience. They are essentially local drug lords, shamans in a time when shamanism is forbidden by the state. The liquor is an ironic symbol for spiritual freedom and the shortcomings of the American government and its tendency to control.
The school
One should think that with a title like The Education of Little Tree, the school might be an important part of the book, but actually, it is more like a prison. The school has no part in Little Tree's true education, it seems, because it is ironically devoid of any "true" knowledge. They teach the kinds of knowledge that can be exploited to earn money, but they neglect to teach the children about their identity, about the mystery of life and death, the nature of heritage and time. For that, Little Tree finds his family is a better source of education.
Christian missionaries
There is an obvious irony concerning the Christian missionaries who are invited into the Native schools, quite literally to brainwash the kids out of their own heritage and into another heritage. The question of the irony is this: Did Jesus Christ really have manipulation, shame, and brainwashing in mind when he offered his gospel to the world? These Christian missionaries are clearly acting like the Pharisees whom Jesus preached against. In other words, the Christian faith as experienced by Little Tree is defined by the very religious closed-mindedness and control that Jesus sought to eliminate. The missionaries are religious hypocrites.
The irony of death
When Little Tree's grandparents die, he has only his puppies. Then, they die. This points back to the frantic emotions he was experiencing at the beginning of the novel. How will he survive in the cruel competitions of nature? Who will protect him? The answer is found in the irony of death. He has been educated by his grandparents about the spiritual realm. He believes that their spirits have survived death. Yes, their bodies were like the dogs, but they were also like something else that seemed immortal to Little Tree. If they have that spiritual aspect, so then does he, so their deaths point him toward his own adult identity (a common motif in world literature).
