I wrote an article about Binary Search Trees a few weeks ago. AVL trees are a specialization of Binary Search Trees.

AVL trees (named after their inventors, Adelson-Velskii and Landis) were the first kind of self-balancing tree to be invented, so their implementation is somewhat simple compared to newer self-balancing trees.

This type of tree allows you to perform insertions, deletions and searches in O(log n). This tree keeps track of the heights of all the nodes. Every time one node is inserted or deleted, the balancing factor (difference between the heights of left and right subtree) of its ancestors is checked. If the balancing factor is greater than 1 or lower than -1, then the tree is rebalanced.

Searching in an AVL tree works exactly the same as searching in a regular BST.

Balancing

Before jumping into insertion and deletion we need to explore balancing the tree. If at any moment the balance factor becomes greater than 1 or lower than -1, then we need to rebalance. Let’s look at an example:

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Start with:

    5         | Height of 5 is 2
  /   \       | Height of 4 is 1
4       6     | Height of 6 is 1
              | Balance factor of 5 is 0
Insert 3:

     5         | Height of 5 is 3
   /   \       | Height of 4 is 2
  4     6      | Height of 6 is 1
 /             | Height of 3 is 1
3              | Balance factor of 5 is 1


Insert 2:

       5         | Height of 5 is 4
     /   \       | Height of 4 is 3
    4     6      | Height of 6 is 1
   /             | Height of 3 is 2
  3              | Height of 2 is 1
 /               | Balance factor of 5 is 2. Rebalancing is needed.
2

After rebalancing:

    4
   /  \
  3    5
 /      \
2        6

Now that we roughly know when to rebalance a tree, we need to look at how to do it.

When a tree becomes unbalanced, it has to be balanced by performing what is called a rotation. There are four different rotations that we can use to balance a tree. I’ll use examples with only three nodes for simplicity, but the examples can be generalized. More specifically, once you find the node where the rotation needs to be made, you can make the rotation and disregard the sub-trees of the nodes below.

LL - Left-Left

An LL rotation is necessary when we have a tree that is too heavy on the right:

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5
    4
      \
       5
        \
         6

For the example above, we can imagine that we started by inserting 4, then 5 and finally 6. After the rotation we end with this:

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    5
  /   \
4       6

The steps we followed to get there are:

  • 4’s parent needs to point to 5 now
  • 4 becomes 5’s left child
  • 5’s left child, becomes 4’s right child

Those are all the changes needed. Everything else stays the same. Let’s add some markers to make this clearer:

This:

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    P
    |
    4
      \
       5
      /  \
     L     6

Becomes this:

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4
5
6
7
    P
    |
    5
  /   \
4       6
 \
  L

RR - Right-Right

This is a mirror of LL:

This:

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7
    P
    |
    6
   /
  5
 / \
4   R

Becomes this:

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4
5
6
7
    P
    |
    5
  /   \
4       6
       /
      R

The steps we followed are:

  • 6’s parent to point to 5 now
  • 6 becomes 5’s right child
  • 5’s right child, becomes 6’s left child

LR - Left-Right

This is where things get a little more interesting. In some situations a single rotation is not enough:

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5
    4
      \
       6
      /
     5

We could try an LL, but that doesn’t really solve anything:

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4
5
    6
   /
  4
   \
    5

The way we solve this is by doing a right rotation on the right sub-tree:

This is where we started:

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3
4
5
    4
      \
       6
      /
     5

This is the right sub-tree of 4:

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2
3
  6
 /
5

After right rotating, this is the tree we get:

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2
3
5
 \
  6

And the whole thing looks like this now:

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3
4
5
4
  \
   5
    \
     6

From here we can do our left rotation, and we’ll have a balanced tree.

RL - Right-Left

This is a mirror or LR. The process looks something like this:

This would be the starting point:

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4
5
  6
 /
4
 \
  5

This is the left sub-tree of 6:

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2
3
4
 \
  5

After left rotating this tree we get:

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3
  5
 /
4

And the whole thing looks like this:

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4
5
    6
   /
  5
 /
4

Balance factor

In order to decide when to rotate we need ot figure out the balance factors of the nodes. The formula to calculate the balance factor is:

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balance factor = height of left subtree - height of right subtree

Calculating the heights of the sub-trees of the root node would require us to traverse the whole tree, which would be very inefficient. For this reason a better approach is to store and update the height of the nodes every time it is needed.

In the Binary Search Trees article we have a _createNode function. Let’s extend it so it includes the height:

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_createNode(val, parent = null) {
  return {
    val: val,
    parent: parent,
    left: null,
    right: null,
    height: 1
  }
}

This way, every time a node is inserted or deleted we don’t have to calculate all heights, since they will be stored with the node itself

Inserting

Let’s look at an example of an insertion. Let’s say we have this tree:

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3
  5
 / \
4   6

If we insert a 7 here, we end with this:

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4
5
  5
 / \
4   6
     \
      7

Besides doing the insertion of the new node, we also need to update the heights and rebalance the tree if necessary. The algorigthm goes like this:

Steps to insert 7:

  • Search where 7 will be inserted (In this case, as the right child of 6)
  • Create a new node (_createNode(7, parentNode))
  • Set 6’s right node to the new node (parentNode.right = newNode)
  • Update 6, and all its ancesters’ heights if necessary
  • Rebalance 6 and all its ancesters if necessary

Deleting

Deleting a node follows the same steps as with a normal Binary Search Tree. You can check my Binary Search Trees article for the instructions. The only difference is that heights of the ancestors of the deleted node might need to be updated, and they might need to be rebalanced.

Code

This code is an extension of the code from my Binary Search Trees article. If there are parts that are not clear, you might want to check it.

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class AvlTree {
  // Create a node
  _createNode(val, parent = null) {
    return {
      val: val,
      parent: parent,
      left: null,
      right: null,
      height: 1
    }
  }

  // Perform a left rotation on the given node
  _leftRotate(node) {
    const nodeRight = node.right;

    if (node.parent) {
      if (node.parent.left === node) {
        node.parent.left = nodeRight;
      } else if (node.parent.right === node) {
        node.parent.right = nodeRight;
      } else {
        console.error('There was an error rotating:');
        console.log(node);
      }
    }
    nodeRight.parent = node.parent;

    node.right = nodeRight.left;
    if (nodeRight.left) {
      nodeRight.left.parent = node.right;
    }

    nodeRight.left = node;
    node.parent = nodeRight;

    // Update the heights
    let rh = node.right ? node.right.height : 0;
    let lh = node.left ? node.left.height : 0;
    node.height = Math.max(rh, lh) + 1;

    let rrh = nodeRight.right ? nodeRight.right.height : 0;
    let rlh = nodeRight.left ? nodeRight.left.height : 0;
    nodeRight.height = Math.max(rrh, rlh) + 1;
  }

  // Perform a right rotation on the given node
  _rightRotate(node) {
    const nodeLeft = node.left;

    if (node.parent) {
      if (node.parent.left === node) {
        node.parent.left = nodeLeft;
      } else if (node.parent.right === node) {
        node.parent.right = nodeLeft;
      } else {
        console.error('There was an error rotating ' + node);
      }
    }
    nodeLeft.parent = node.parent;

    node.left = nodeLeft.right;
    if (nodeLeft.right) {
      nodeLeft.right.parent = node.left;
    }

    nodeLeft.right = node;
    node.parent = nodeLeft;

    // Update the heights
    let rh = node.right ? node.right.height : 0;
    let lh = node.left ? node.left.height : 0;
    node.height = Math.max(rh, lh) + 1;
    let lrh = nodeLeft.right ? nodeLeft.right.height : 0;
    let llh = nodeLeft.left ? nodeLeft.left.height : 0;
    nodeLeft.height = Math.max(lrh, llh) + 1;
  }

  // Search for a value in this tree. If not found, returns
  // the closest node (the node where the value would be
  // inserted)
  _findPosition(val) {
    let current = this._root;
    while (true) {
      if (current.val === val) {
        return current;
      }

      if (val > current.val && current.right) {
        // The value we are searching for is greater.
        // Continue on the right subtree
        current = current.right;
      } else if (val < current.val && current.left) {
        // The value we are searching for is lower.
        // Continue on the left subtree
        current = current.left;
      } else {
        // No more tree to traverse
        return current;
      }
    }
  }

  // Updates heights of the node and all its ancesters, based on the
  // height of its children
  _updateHeights(node) {
    for (let current = node; current; current = current.parent) {
      let rh = current.right ? current.right.height : 0;
      let lh = current.left ? current.left.height : 0;
      current.height = Math.max(rh, lh) + 1;
    }
  }

  // Balances the tree if necessary. Updates the _root accordingly
  _balance(node) {
    let current = node;
    let last = current;
    while (current) {
      let leftHeight = current.left ? current.left.height : 0;
      let rightHeight = current.right ? current.right.height : 0;
      let factor = leftHeight - rightHeight;

      // LL or LR needed
      if (factor < -1) {
        let currentRight = current.right;
        let lh = currentRight.left ? currentRight.left.height : 0;
        let rh = currentRight.right ? currentRight.right.height : 0;
        let diff = lh - rh;

        if (diff === 1) {
          // LR
          this._rightRotate(current.right);
          this._leftRotate(current);
        } else {
          // LL
          this._leftRotate(current);
        }
      }

      // RR or RL needed
      if (factor > 1) {
        let currentLeft = current.left;
        let lh = currentLeft.left ? currentLeft.left.height : 0;
        let rh = currentLeft.right ? currentLeft.right.height : 0;
        let diff = lh - rh;

        if (diff === -1) {
          // RL
          this._leftRotate(current.left);
          this._rightRotate(current);
        } else {
          // RR
          this._rightRotate(current);
        }
      }

      last = current;
      current = current.parent;
    }

    this._root = last;
  }

  // Deletes the given node. The node must have at most
  // one child
  _deleteSimple(node) {
    if (!node.left && !node.right) {
      // No children, simple delete
      if (!node.parent) {
        this._root = null;
      } else if (node.parent.right == node) {
        node.parent.right = null;
      } else {
        node.parent.left = null;
      }
    } else {
      // Single child, link the child to the parent
      if (!node.parent) {
        if (node.left) {
          this._root = node.left;
          node.left.parent = node.parent;
        } else {
          this._root = node.right;
          node.right.parent = node.parent;
        }
      } else if (node.parent.right == node) {
        if (node.left) {
          node.left.parent = node.parent;
          node.parent.right = node.left;
        } else {
          node.right.parent = node.parent;
          node.parent.right = node.right;
        }
      } else {
        if (node.left) {
          node.left.parent = node.parent;
          node.parent.left = node.left;
        } else {
          node.right.parent = node.parent;
          node.parent.left = node.right;
        }
      }
    }
  }

  // Search for a value in this tree
  search(val) {
    const found = this._findPosition(val);
    if (found && found.val === val) {
      return found;
    }
  }

  // insert a value
  insert(val) {
    // The case where this is the first value
    if (!this._root) {
      this._root = this._createNode(val);
      return;
    }

    // Search for the value. If it is found we don't need
    // to do anything
    let position = this._findPosition(val);
    if (position.val === val) {
      return;
    }

    // Insert the new value in the tree
    if (position.val > val) {
      position.left = this._createNode(val, position);
    } else {
      position.right = this._createNode(val, position);
    }

    // For an AVL tree we need to update the heights and balance
    this._updateHeights(position);
    this._balance(position);
  }

  // Delete a value from the tree
  delete(val) {
    // Search for the value. If it is not found we don't
    // need to do anything
    let node = this.search(val);
    if (!node) {
      return;
    }

    let deleteParent = null;
    if (node.left && node.right) {
      // Node has two children
      let lowestRightNode = node.right;
      while (lowestRightNode.left) {
        lowestRightNode = lowestRightNode.left;
      }

      let tempVal = lowestRightNode.val;
      lowestRightNode.val = node.val;
      node.val = tempVal;
      this._deleteSimple(lowestRightNode);
      deleteParent = lowestRightNode.parent;
    } else {
      this._deleteSimple(node);
      deleteParent = node.parent;
    }

    // For an AVL tree we need to update the heights and balance
    this._updateHeights(deleteParent);
    this._balance(deleteParent);
  }
}

// Create an AVL tree
let avl = new AvlTree();

// Test insertions and balancing:
avl.insert(1);
avl.insert(2);
avl.insert(3);
avl.insert(4);
avl.insert(5);
avl.insert(6);

// Test deletions and balancing:
avl.delete(6);
avl.delete(5);
console.log(avl._root);
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