Who is the narrator, can she or he read minds, and, more importantly, can we trust her or him?
Third Person (Omniscient)
There's no "I" in "tree" or "boy," and we definitely don't have a first-person narrator here. Instead, we get the story from the perspective of a nameless, faceless onlooker who is aware of how the tree and the boy feel and knows what happens in their relationship over the course of 70 or so years. So, yes, this story definitely features a third-person omniscient POV.
What's interesting about this particular use of third-person omniscient POV is that the narrator seems to lose touch with the boy over the course of the story just as the tree does. In the beginning, the narrator tells us straight up that the boy loves the tree. But that's the only peek we get directly inside the boy's head.
For the rest of the story, we get the boy's feelings and needs from his dialogue. "I'm too big to climb and play" (36), he says. Later, he tells us, "I'm too old and sad to play" (46).
The narrator presumably knows these things about the boy but chooses to communicate them through dialogue instead of direct telling. The tree's feelings, on the other hand, continue to be communicated by the narrator. We are told that "the tree was happy" on five different occasions, although on page 51, we get the qualifier "…but not really."
Of course, The Giving Tree is a picture book, which means that some aspects of the story are communicated visually instead of through the text. In that way, we can see that the narrator does indeed still have a connection to the boy's inner world. How? From the drawings that so clearly depict his emotions. On pages 37 and 41, the boy gazes longingly at the tree, first asking for money and then for a house.
The sparse illustrations show the boy's resigned posture, and we can see that he is dissatisfied in life, lacking something, even though he doesn't seem to know what. And later, on pages 47 and 52, his stooped back and grizzled visage communicate his bitterness and lack of fulfillment as well. So, even though the narrator shies away from directly telling us the boy's state of mind, it's clear that our omniscient narrator still knows exactly how the boy feels.
One more thing about the POV: since some people interpret the relationship between the boy and the tree as an abusive one, there could also be a question about the narrator's reliability.
If, for instance, we view the tree as a victim of domestic violence—or perhaps as nature, being consumed by humanity—we have to wonder at the repeated refrain of "and the tree was happy." Is the tree really happy? Is the narrator being facetious? Is the narrator perhaps not omniscient and instead being deceived by a tree who keeps insisting that she's happy when really she's suffering from battered-apple-tree syndrome?
These are all great questions for discussion if you're studying this book for a children's literature class or wondering just how complex and disturbing Shel Silverstein meant for this picture book to be. But if you're reading it to a young person as a bedtime story, you might want to leave that part of the discussion for future years.