There's no doubt about it: this is one very giving tree. So much so, she deserves to have the book named after her.
Indeed, the eponymous main character in The Giving Tree gives up her apples, all her limbs, and her entire trunk for the love of her boy. She gives until there's nothing left to give, and then she gives that, too. And she does it all without ever expecting anything in return.
Now that's giving.