The AVL tree is a self-balancing binary search tree, introduced in 1962 by GM Adelson-Velsky and EM Landis in their paper "An algorithm for the organization of information." This data structure ensures that the height difference between the left and right subtrees of any node does not exceed one. The height of an empty node is considered 0, while the height of a leaf node is 1.
Unlike a regular binary search tree, which can degenerate into a linked list when inserting ordered data, an AVL tree maintains a balanced structure, ensuring logarithmic time complexity for search operations. This makes it significantly more efficient for large datasets.
When inserting or deleting nodes, the AVL tree may become unbalanced. To restore balance, rotations are performed on the affected subtree. There are four types of imbalances: left-left, right-right, left-right, and right-left. Each imbalance requires a specific rotation to correct the structure.
For example, a left-left imbalance is corrected with a left-left rotation (LL rotation), where the left child becomes the new root of the subtree. Similarly, a right-right imbalance is resolved using a right-right rotation (RR rotation). For more complex imbalances like left-right or right-left, two rotations are needed—first a single rotation on the child, followed by a second rotation on the parent node.
Here’s a simplified code snippet showing how the LL rotation works:
Node * AVL::ll_rotate(Node * y) {
Node * x = y->left;
y->left = x->right;
x->right = y;
y->height = max(get_height(y->left), get_height(y->right)) + 1;
x->height = max(get_height(x->left), get_height(x->right)) + 1;
return x;
}
The same logic applies to other rotations, with variations depending on the type of imbalance. After each insertion or deletion, the tree checks for balance and performs the necessary rotations to maintain its properties.
The implementation also includes functions for searching and deleting nodes. The search operation follows the standard binary search tree approach, while deletion involves finding a successor node and replacing the deleted node with it before rebalancing the tree if needed.
One important point to note is that after performing a rotation, the height of the affected nodes must be updated to ensure future balance checks are accurate. This is crucial for maintaining the integrity of the AVL tree.
Overall, the AVL tree provides a reliable and efficient way to manage dynamic datasets with fast search, insert, and delete operations. While it has some overhead due to balancing, its performance guarantees make it a popular choice in many applications.
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