Today we are going to discuss:
- What is a tree?
- Basic tree terminologies
- Types of trees
- Operations on trees
- Some practice questions
🌳 What is a tree?
Trees are hierarchical data structures used to organize and represent data in a manner that resembles a tree with a root at the top and branches connecting nodes. Each node can have child nodes, creating a parent-child relationship.
Tree data structures have various real-world applications:
- File Explorer: Trees organize files and folders hierarchically, making it easy to navigate and locate data on your device.
- Parsers (XML parser): Trees structure and interpret data from files like XML, allowing software to understand and process information effectively.
- Database for Indexing: Trees enable efficient data storage and retrieval in databases, speeding up searches.
- DNS (Domain Name System): Trees convert domain names into IP addresses, facilitating internet communication.
- Trie (Autocomplete): Trees power autocomplete suggestions in search engines by quickly searching for and suggesting relevant terms or phrases as you type.
🌳 Basic tree terminologies
- Parent: A node in a tree data structure that has one or more child nodes directly connected to it.
- Child: A node in a tree data structure that is connected to a parent node.
- Root: The topmost node in a tree, serving as the starting point for traversing the structure. It has no parent.
- Leaf: A node in a tree that has no children, meaning it is a terminal node.
- Internal Node: A node in a tree that is not a leaf, which means it has one or more children.
- Level: The position of a node within a tree, with the root node at level 0 and each subsequent level increasing by 1.
- Sibling: Nodes that share the same parent in a tree.
- Degree of a Node: The number of subtrees (children) connected to a node. Leaf nodes have a degree of 0.
- Height of a Tree: The length of the longest path from the root to a leaf node in the tree.
- Height of a Node: The length of the longest path from the node to a leaf node in the tree.
- Depth of a Node: The length of the path from the root to that node, measured in terms of the number of edges in the path.
🌳 Types of trees
Now that we know what a tree is, let use look into a few different types of trees. You must have came across a binary tree or a binary search tree.
General Tree: A tree structure where nodes can have any number of children, providing flexibility in organising hierarchical data.
Binary Tree: A tree in which each node can have at most two children, commonly used for efficient data organisation and traversal.
Binary Search Tree (BST): A binary tree where values are organised so that left children are smaller, and right children are equal to or greater than the parent, enabling efficient searching and sorting.
AVL Trees: A self-balancing binary search tree that ensures the height difference between left and right subtrees is kept to a minimum, guaranteeing efficient operations.
Red-Black Trees: Another self-balancing binary search tree that maintains balance by using colored nodes and performing rotations, suitable for a variety of applications.
N-ary Trees: Trees where nodes can have more than two child nodes, allowing for versatile data representation.
B-Trees: Self-balancing trees used for managing large datasets, frequently employed in databases and file systems.
B+ Trees: A variant of B-trees with enhanced properties, particularly useful for indexing large amounts of data in databases.
Trie (Prefix Tree): A tree structure optimized for storing and searching strings, commonly employed in applications such as autocomplete and spell-checking.
Ternary Tree: A tree with nodes that have exactly three children, often used in specialised scenarios requiring a three-way decision structure.
Segment Tree: A binary tree structure designed for efficient range queries, especially useful for finding intervals or segments that contain a given query point, offering fast retrieval in O(log n + k) time, where k represents the number of retrieved intervals or segments.
🌳 Operations on trees
We can perform a variety of operations on trees. Let us have a look on some of the common ones.
- Insertion: Add a new node with a given value to the tree.
- Deletion: Remove a specific node (and its subtree) from the tree.
- Search: Find a specific node with a given value in the tree.
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Traversal:
- Inorder: Visit nodes in sorted order (for binary search trees).
- Preorder: Visit the current node before its children.
- Postorder: Visit the current node after its children.
- Level-order: Visit nodes level by level, starting from the root.
- Find Minimum/Maximum: Find the node with the smallest or largest value in the tree.
- Balancing: Rebalance the tree to ensure that it maintains a specific structure, such as in AVL trees or Red-Black trees.
- Height Calculation: Calculate and return the height (or depth) of the tree.
- Count Nodes: Count and return the number of nodes in the tree.
- Diameter Calculation: Find the length of the longest path between any two nodes in the tree.
- Lowest Common Ancestor (LCA): Find the lowest common ancestor of two nodes in the tree.
- Checking for Balance: Verify whether the tree is balanced, ensuring the heights of subtrees do not differ by more than a certain threshold.
- Serialization and Deserialization: Convert the tree into a serial format for storage or transmission, and then reconstruct the tree from the serialized data.
- Prefix Search (in Tries): Search for words or strings based on a common prefix in a Trie data structure.
- Traversal for Specific Orderings: Perform various tree traversals (e.g., in-order, preorder, postorder) with specific requirements or rules for processing nodes.
- Pruning: Remove specific nodes or subtrees from the tree based on certain conditions or criteria.
🌳 Some practice questions
To understand trees better and develop an intuition for solving tree problems, I recommend the following problems.
Before anything else, Implement a simple BST with the functions to create node, insert node, delete node and tree-traversal, after that do the following questions:
- Invert Binary Tree
- Lowest Common Ancestor of a Binary Search Tree
- Maximum Depth of Binary Tree
- Balanced Binary Tree
- Diameter of Binary Tree
- Binary Tree Level Order Traversal
- Implement Trie (Prefix Tree)
- Validate Binary Search Tree
- Lowest Common Ancestor of a Binary Tree
- Binary Tree Right Side View
- Serialize and Deserialize Binary Tree
- Construct Binary Tree from Preorder and Inorder Traversal
- Minimum Height Trees
- Kth Smallest Element in a BST
Links -
- More questions relevant to BST :
- Run your c++ code online : https://ideone.com/
- Structures in C++
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