Balanced Binary Search Trees: A Deep Dive with JavaScript
Balanced Binary Search Trees (BBST) are a unique blend of binary search trees and balanced trees, making them integral in both theoretical and practical aspects of computer science. Let’s embark on a journey to understand these trees and their operations, bolstered with JavaScript examples.
Balanced Binary Search Trees: A Definition
A BBST is a binary search tree that ensures the tree remains balanced after every insert or delete operation. In essence, this ‘balance’ ensures that operations such as search, insert, and delete always run in logarithmic time, making them efficient and predictable.
The Need for Balance
While binary search trees offer a structured way of storing data, their performance can degrade in skewed or unbalanced scenarios. For instance, if you insert elements in sorted order, the tree becomes a linked list, leading to operations taking linear time. BBSTs remedy this by ensuring the tree remains balanced, thus guaranteeing logarithmic time complexities.
Common Balanced Binary Search Trees:
AVL Tree:
Named after Adelson-Velsky and Landis, the AVL tree ensures the height difference between the left and right children of any node is no more than one. Let’s see an insertion example in JavaScript:
class Node {
constructor(key) {
this.key = key;
this.left = null;
this.right = null;
this.height = 1; // height of the node
}
}
class AVLTree {
// ... other utility functions
// Function to insert a key in AVL Tree
insert(root, key) {
// Regular BST insertion
if (!root) return new Node(key);
if (key < root.key) root.left = this.insert(root.left, key);
else if (key > root.key) root.right = this.insert(root.right, key);
else return root; // Duplicates not allowed
// Update height
root.height = 1 + Math.max(this.getHeight(root.left), this.getHeight(root.right));
// Check balance
let balance = this.getBalance(root);
// Rotations (4 cases)
// Left Left
if (balance > 1 && key < root.left.key) return this.rightRotate(root);
// Right Right
if (balance < -1 && key > root.right.key) return this.leftRotate(root);
// Left Right
if (balance > 1 && key > root.left.key) {
root.left = this.leftRotate(root.left);
return this.rightRotate(root);
}
// Right Left
if (balance < -1 && key < root.right.key) {
root.right = this.rightRotate(root.right);
return this.leftRotate(root);
}
return root;
}
}
Red-Black Tree:
Red-Black Trees color each node either red or black and then maintain certain properties after every insertion or deletion to ensure balance.
Splay Tree:
These are self-adjusting trees. After operations, they move the accessed key to the root, ensuring a splaying effect that balances the tree over time.
B-Trees:
Known for their use in databases, B-trees can store more than one item at each node and maintain a balanced structure.
Operations and Their Importance:
- Insertion: BBSTs may employ tree rotations after inserting a node to restore balance. For instance, AVL trees use four types of rotations.
- Deletion: Similar to insertion but often more complex. Balancing post deletion is essential to ensure the tree’s efficiency.
- Lookup: BBSTs leverage their balanced nature to provide faster lookups than regular BSTs.
Conclusion:
Balanced Binary Search Trees are versatile and crucial in various domains. While they come with an overhead of maintaining balance, the consistency and efficiency they offer make them a valuable asset in data structure and algorithm toolkits, especially when implemented in languages like JavaScript.