The AVL tree is a self-balancing binary search tree, introduced by G.M. Adelson-Velsky and E.M. Landis in their 1962 paper "An algorithm for the organization of information." The key feature of an AVL tree is that it maintains balance, ensuring that the height difference between the left and right subtrees of any node does not exceed one. This height convention defines the height of an empty node as 0 and a leaf node as 1.
In contrast to a regular binary search tree, which can degenerate into a linked list when inserting ordered data, an AVL tree ensures logarithmic time complexity for operations like insertion, deletion, and search. This makes it significantly more efficient for large datasets where performance is critical.
When inserting or deleting nodes, the 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 requires a specific rotation strategy to rebalance the tree.
For example, a left-left imbalance occurs when the left subtree of a node is two levels taller than the right subtree, and its left child's left subtree is one level taller than its right. In this case, a single right rotation (LL rotation) restores balance. Similarly, other imbalances require different rotation techniques such as LR (left-right), RL (right-left), and RR (right-right).
The implementation involves recursive functions that update the height of each node after insertions or deletions and check for imbalances. If an imbalance is detected, the appropriate rotation is applied, and the new root of the subtree is returned to maintain the structure of the entire tree.
Here is a simplified version of the code for inserting a node into an AVL tree:
Node * AVL::insert_real(int key, Node * node) {
if (node == nullptr) return new Node(key);
if (key < node->key)
node->left = insert_real(key, node->left);
else if (key > node->key)
node->right = insert_real(key, node->right);
else
return node;
node->height = max(get_height(node->left), get_height(node->right)) + 1;
int balance = get_balance(node);
if (balance > 1 && get_balance(node->left) > 0)
return ll_rotate(node);
if (balance < -1 && get_balance(node->right) < 0)
return rr_rotate(node);
if (balance > 1 && get_balance(node->left) < 0)
return lr_rotate(node);
if (balance < -1 && get_balance(node->right) > 0)
return rl_rotate(node);
return node;
}
Similarly, the deletion process follows a similar approach but involves additional steps to handle cases where a node has two children. The successor of the node is found, copied, and then deleted from the right subtree, ensuring the tree remains balanced throughout the operation.
The main advantage of an AVL tree over a regular binary search tree is its guaranteed logarithmic time complexity for all operations, making it ideal for applications requiring fast data retrieval and modification. However, the overhead of maintaining balance through rotations can make it slightly slower than other balanced trees like Red-Black Trees in certain scenarios.
UL XLPE Cable,UL3385 electronic wire,High-temperature resistant wires,Irradiation cross-linked wires
Jiangyin City Weicheng Special Cable Co.,Ltd , https://www.weichengcable.com