Introduction to Trees in Data Structures

In Data Structures and Algorithms (DSA), trees are a foundational data structure with hierarchical structures, commonly used for representing data that has a parent-child relationship. Learn Introduction to Trees in Data Structures, including types, properties & applications, essential for mastering hierarchical data organization in computing Here’s an overview of trees in DSA:

1. Basic Terminology

• Node: Each element in a tree.
• Root: The top node of a tree.
• Parent and Child: Direct relationships between nodes. A parent node has child nodes connected beneath it.
• Leaf Node: A node with no children.
• Edge: The connection between two nodes.
• Height: The length of the longest path from the root to a leaf.
• Depth: The distance from the root to a specific node.
• Subtree: A smaller tree derived from a larger tree starting at a node.

 

2. Types of Trees

• Binary Tree: Each node has at most two children, typically known as the left and right child.
• Binary Search Tree (BST): A binary tree with ordered nodes—values on the left are smaller than the parent, and values on the right are larger.
• Balanced Tree: Trees like AVL and Red-Black Trees, are balanced to keep operations efficient.
• Heaps: A complete binary tree used in priority queues, where the parent node is either always greater than (Max Heap) or smaller than (Min Heap) its children.
• Trie: Used to store strings, commonly in text processing or autocomplete functionalities.
• B-Trees and B+ Trees: Used in databases for storing data in a way that minimizes disk reads.

 

3. Common Operations

• Insertion: Adding a new node.
• Deletion: Removing a node.
• Traversal: Visiting nodes in a specific order, including:
  • In-order (left, root, right): Used in BSTs to get sorted order.
  • Pre-order (root, left, right): Used to replicate the tree structure.
  • Post-order (left, right, root): Used to delete or free nodes.
  • Level-order: Visits nodes level by level (useful for BFS).

 

4. Applications

• Binary Search Trees: Used in searching and sorting.
• Heaps: Useful in implementing priority queues.
• Tries: Often used for autocomplete systems, spell checking, and IP routing.
• B-Trees and B+ Trees: Essential in database indexing.

 

5. Complexities

• Tree operations typically vary based on the type of tree. For instance:
  • BST Search, Insert, Delete: O(log n)O(\log n)O(logn) if balanced, but O(n)O(n)O(n) if skewed.
  • Traversal: Generally O(n)O(n)O(n) for all types.

 

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Operations on trees involve a set of fundamental actions for managing and interacting with tree structures. Here’s a closer look at the primary operations performed on trees:

1. Traversal

Tree traversal is the process of visiting each node in a tree in a specific order. The main types of tree traversal are:

• In-order Traversal (Left, Root, Right):
  • Commonly used in Binary Search Trees (BST) to retrieve data in sorted order.
  • For a BST, this traversal will visit nodes in ascending order.
• Pre-order Traversal (Root, Left, Right):
  • Used for creating a copy of the tree and for prefix expressions in mathematical trees.
• Post-order Traversal (Left, Right, Root):
  • Useful for deleting or freeing nodes from memory.
  • Post-order is also used in evaluating postfix expressions.
• Level-order Traversal (Breadth-First Traversal):
  • Visits nodes level by level, from the root down.
  • This traversal is implemented using a queue and is useful for finding the shortest path in unweighted trees.

 

2. Insertion

• Binary Trees: In a generic binary tree, a new node is usually added to the first available position, maintaining the complete tree property.
• Binary Search Trees (BSTs): New nodes are added based on their value; smaller values go to the left, and larger values go to the right.
• AVL Trees and Red-Black Trees: Insertion involves additional steps to keep the tree balanced, as well as adjusting node heights or colors to maintain balanced properties.

 

3. Deletion

Deletion in trees can be complex, as it often requires restructuring the tree to maintain its properties:

• Binary Search Trees (BSTs):
  • Deleting a leaf node is straightforward, simply removing it.
  • Deleting a node with one child involves connecting the child directly to the parent.
  • Deleting a node with two children requires finding either the in-order successor (smallest node in the right subtree) or the in-order predecessor (largest node in the left subtree) and replacing the node to be deleted with it.
• Balanced Trees: In AVL and Red-Black trees, deletion may also require additional rotations to keep the tree balanced.

 

4. Searching

• Binary Search Trees (BSTs): Searching is efficient due to the ordered property, with an average time complexity of O(log n)O(\log n)O(logn) for balanced trees.
• Tries: In trie structures, searches are conducted character-by-character, making them ideal for string matching.

 

5. Balancing

• Trees like AVL and Red-Black Trees implement balancing to keep operations efficient.
• Rotations are used to restore balance after insertions or deletions.
  • Right Rotation: Moves a left-heavy subtree up and the root down.
  • Left Rotation: Moves a right-heavy subtree up and the root down.

 

6. Finding Minimum and Maximum

• In a BST, the minimum value is found by traversing to the leftmost node, while the maximum is found by traversing to the rightmost node.
• In general trees, these operations may require a full traversal to compare all nodes.

 

7. Height or Depth Calculation

• The height of a tree is the longest path from the root to any leaf node.
• The depth of a node is the distance from the root to that node.
• Calculating height and depth can help assess tree balance and optimize search performance.

 

8. Lowest Common Ancestor (LCA)

• Finding the LCA of two nodes is a common operation, especially in binary trees and BSTs.
• LCA is the lowest node that has both given nodes as descendants.

 

9. Merging Trees

• Merging two binary trees typically combines nodes with the same position in both trees.
• Merging AVL or Red-Black Trees may involve creating a new balanced tree.

 

Summary of Complexities:

Operation	BST Average	Balanced Tree (AVL/Red-Black)
Insert	O(log n)O(\log n)O(log n)	O(log n)O(\log n)O(log n)
Delete	O(log n)O(\log n)O(log n)	O(log n)O(\log n)O(log  n)
Search	O(log n)O(\log n)O(log n)	O(log n)O(\log n)O(log n)
Traversal	O(n)O(n)O(n)	O(n)O(n)O(n)

 

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Author:-

Sarika Ganesh Kore

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