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Commit 3786c16

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avl tree docs
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‎book/chapters/tree--avl.adoc‎

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= AVL Tree
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AVL Tree is named after their inventors (**A**delson-**V**elsky and **L**andis).
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This self-balancing tree keep track of subtree sizes to know if a rebalance is needed or not.
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We can compare the size of the left and right subtrees using a balance factor.
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[NOTE]
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====
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The *balanced factor* on each node is calculated recurviely as follows:
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----
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Balance Factor = (left subtree height) - (right subtree height)
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----
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====
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The implementation will got into the BST node.
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We will need two methods to calculate the left and right subtree, and with those we can get the balance factor.
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.Balance Factor methods on the BST node
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[source, javascript]
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----
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include::{codedir}/data-structures/trees/binary-tree-node.js[tag=avl, indent=0]
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----
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== Implementing AVL Tree
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Implementing an AVL Tree is not too hard, since it builds upon what we did in the Binary Search Tree.
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.AVL Tree class
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[source, javascript]
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----
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include::{codedir}/data-structures/trees/avl-tree.js[tag=AvlTree]
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----
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As you can see, AVL tree inherits from the BST class.
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The insert and remove operations works the same as in the BST, except that at the end we call `balanceUptream`.
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This function makes balance the tree after every change if is needed. Let's see how it's implemented.
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.Balance Upstream for AVL tree
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[source, javascript]
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----
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include::{codedir}/data-structures/trees/avl-tree.js[tag=balanceUptream]
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----
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This function recurively goes from the modified node to the root checking if each node in between is balanced.
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Now, let's examine how does the balancing works on AVL tree.
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.Balance method for AVL tree
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[source, javascript]
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----
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include::{codedir}/data-structures/trees/avl-tree.js[tag=balance]
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----
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The first thing we do is to see if one subtree is longer than the other.
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If so, then we check the children balance to determine if need a single or double rotation and in which direction.
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You can review <<Tree Roations>> in case you want a refresher.

‎src/data-structures/trees/avl-tree.js‎

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class AvlTree extends BinarySearchTree {
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/**
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* Add node to tree. It self-balance itself.
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*
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* @param {any} value node's value
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*/
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add(value) {

‎src/data-structures/trees/binary-tree-node.js‎

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this.meta.color = value;
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}
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// tag::avl[]
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/**
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* @returns {Number} left subtree height or 0 if no left child
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*/
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get leftSubtreeHeight() {
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return this.left ? this.left.height + 1 : 0;
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}
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/**
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* @returns {Number} right subtree height or 0 if no right child
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*/
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get rightSubtreeHeight() {
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return this.right ? this.right.height + 1 : 0;
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}
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/**
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* Get the max height of the subtrees.
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*
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return Math.max(this.leftSubtreeHeight, this.rightSubtreeHeight);
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}
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get leftSubtreeHeight() {
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return this.left ? this.left.height + 1 : 0;
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}
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get rightSubtreeHeight() {
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return this.right ? this.right.height + 1 : 0;
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}
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/**
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* Returns the difference the heights on the left and right subtrees
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*/
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get balanceFactor() {
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return this.leftSubtreeHeight - this.rightSubtreeHeight;
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}
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// end::avl[]
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/**
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* Serialize node's values

‎src/data-structures/trees/tree-rotations.js‎

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* /
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* 1
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*
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* @param {TreeNode} node this is the node we want to rotate to the right. (E.g., node 3)
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* @param {TreeNode} node
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* this is the node we want to rotate to the right. (E.g., node 3)
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* @returns {TreeNode} new parent after the rotation (E.g., node 2)
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*/
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function rightRotation(node) {
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*
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* @example LR rotation on node 3
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* 4 4
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* / / 4
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* 3 3* /
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* / / 2
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* 1* --left-rotation(1)-> 2 --right-rotation(3)-> / \
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* \ / 1 3*
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* / / 4
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* 3 3* /
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* / / 2
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* 1* --left-rotation(1)-> 2 --right-rotation(3)-> / \
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* \ / 1 3*
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* 2 1
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*
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* @param {TreeNode} node this is the node we want to rotate to the right. E.g., node 3
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* @param {TreeNode} node
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* this is the node we want to rotate to the right. E.g., node 3
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* @returns {TreeNode} new parent after the rotation
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*/
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function leftRightRotation(node) {
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* @example RL rotation on 1
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*
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* 1* 1*
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* \ \ 2
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* 3 -right-rotation(3)-> 2 -left-rotation(1)-> / \
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* / \ 1* 3
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* \ \ 2
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* 3 -right-rotation(3)-> 2 -left-rotation(1)-> / \
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* / \ 1* 3
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* 2 3
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*
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* @param {TreeNode} node

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