Binary Search Tree Basics

Binary Search Tree (BST)

• Definition:

  • A binary search tree is a data structure that maintains sorted order, allowing for efficient searching, insertion, and deletion of elements.

• Properties:

  • Each node has at most two children.
  • The left child contains values less than the parent node.
  • The right child contains values greater than the parent node.
  • The left and right subtrees are also binary search trees.

• Basic Operations:

  1. Searching:

     • Start at the root.
     • Compare the target value with the current node’s value.
     • Move left if the target is smaller, right if larger.
     • Repeat until the target is found or a leaf node is reached.

  2. Insertion:

     • Start at the root and find the appropriate null position where the new value can be placed.
     • Follow the same logic as searching to find the correct location.

  3. Deletion:

     • Three cases to consider:
       • Leaf Node: Simply remove the node.
       • One Child: Remove the node and replace it with its child.
       • Two Children: Find the in-order predecessor (max of left subtree) or in-order successor (min of right subtree), replace the node with it, and delete the predecessor/successor.

• Traversal Methods:

  1. In-order (Left, Node, Right): Yields sorted order of elements.
  2. Pre-order (Node, Left, Right): Useful for copying the tree.
  3. Post-order (Left, Right, Node): Useful for deleting the tree.

• Time Complexity:

  • Average case for search, insertion, and deletion: O(log n).
  • Worst case (unbalanced tree): O(n).

• Balancing:

  • To maintain O(log n) performance, consider self-balancing BSTs like AVL trees or Red-Black trees.

• Applications:

  • Database indexing, memory management, and maintaining a dynamic list of sorted data.

Definition

• A binary search tree (BST) is a data structure that maintains sorted order for efficient searching, insertion, and deletion of elements.

Properties

• Nodes in a BST can have up to two children.
• The left child contains values less than its parent node.
• The right child contains values greater than its parent node.
• Both left and right subtrees must also be binary search trees.

Basic Operations

• Searching:

  • Begin at the root and compare the target value with the current node.
  • Move left if the target is smaller, or right if larger.
  • Continue until the target is found or a leaf node is reached.

• Insertion:

  • Initiate at the root and locate the correct null position for the new value.
  • Employ the same logic as searching to determine where to insert.

• Deletion:

  • Leaf Node: Remove the node directly.
  • One Child: Remove the node and connect its child directly to its parent.
  • Two Children: Replace the node with its in-order predecessor (maximum of the left subtree) or in-order successor (minimum of the right subtree), then delete the predecessor/successor.

Traversal Methods

• In-order (Left, Node, Right): Produces sorted order of elements.
• Pre-order (Node, Left, Right): Useful for creating a copy of the tree.
• Post-order (Left, Right, Node): Effective for deleting the entire tree.

Time Complexity

• Average time complexity for searching, insertion, and deletion is O(log n).
• In the worst-case scenario with an unbalanced tree, time complexity escalates to O(n).

Balancing

• To ensure O(log n) performance, self-balancing BSTs like AVL trees or Red-Black trees are recommended.

Applications

• Commonly used in database indexing.
• Plays a role in memory management.
• Helps maintain a dynamic list of sorted data.

This quiz covers the fundamentals of Binary Search Trees (BST), including their definition, properties, and basic operations like searching, insertion, and deletion. Test your understanding of how BSTs maintain sorted order and efficiently manage data. Perfect for students of data structures!