The tree calls the boy "Boy" even after he grows up and becomes a teenager, a man, a middle-aged man, and an elderly man. This is the tree's way of indirectly saying, "Even if you're too old to spend time with me anymore, I'll always see you as the little boy that used to swing from my branches and eat my apples and nap in my shade."
Alternatively it could be because she is, well, a tree.
The tree calling the boy "Boy" is also a way of pointing out the metaphor for parenthood. We call our offspring our "children" even when they are fully grown. The boy is the Tree's "boy" in the same way a man would be his mother's "boy."
The boy and the tree apparently played hide-and-seek with each other when he was younger, but the tree, who can't move, wouldn't be able to hide or seek properly.
When the boy is a teenager, he picks all the tree's apples to sell, and in the following parts the tree is shown not to have apples anymore. There's no reason that the tree wouldn't have regrown her apples if the boy has been gone for years, unless someone else was also picking them.
When the boy visits the tree as an adult, he asks her to give him a house, and she replies that she has no house to give, because her "house" is the forest she lives in. In other words, she's surrounded by other trees.note (Presumably; we never see them in the book's illustrations.) Out of all these trees, why would the boy choose the tree that was his friend throughout his childhood as the one to cut all the branches off of?
At her insistence, mostly. Note only that he asks for her help; it's her idea to remove the branches for him to build a house.
Presumably for the same reason someone in need of cash asks a friend or family member instead of some random stranger.