Data Structures Tutorial

Data Structures Tutorial Asymptotic Notation Structure and Union Array Data Structure Linked list Data Structure Type of Linked list Advantages and Disadvantages of linked list Queue Data Structure Implementation of Queue Stack Data Structure Implementation of Stack Sorting Insertion sort Quick sort Selection sort Heap sort Merge sort Bucket sort Count sort Radix sort Shell sort Tree Traversal of the binary tree Binary search tree Graph Spanning tree Linear Search Binary Search Hashing Collision Resolution Techniques

Misc Topic:

Priority Queue in Data Structure Deque in Data Structure Difference Between Linear And Non Linear Data Structures Queue Operations In Data Structure About Data Structures Data Structures Algorithms Types of Data Structures Big O Notations Introduction to Arrays Introduction to 1D-Arrays Operations on 1D-Arrays Introduction to 2D-Arrays Operations on 2D-Arrays Strings in Data Structures String Operations Application of 2D array Bubble Sort Insertion Sort Sorting Algorithms What is DFS Algorithm What Is Graph Data Structure What is the difference between Tree and Graph What is the difference between DFS and BFS Bucket Sort Dijkstra’s vs Bellman-Ford Algorithm Linear Queue Data Structure in C Stack Using Array Stack Using Linked List Recursion in Fibonacci Stack vs Array What is Skewed Binary Tree Primitive Data Structure in C Dynamic memory allocation of structure in C Application of Stack in Data Structures Binary Tree in Data Structures Heap Data Structure Recursion - Factorial and Fibonacci What is B tree what is B+ tree Huffman tree in Data Structures Insertion Sort vs Bubble Sort Adding one to the number represented an array of digits Bitwise Operators and their Important Tricks Blowfish algorithm Bubble Sort vs Selection Sort Hashing and its Applications Heap Sort vs Merge Sort Insertion Sort vs Selection Sort Merge Conflicts and ways to handle them Difference between Stack and Queue AVL tree in data structure c++ Bubble sort algorithm using Javascript Buffer overflow attack with examples Find out the area between two concentric circles Lowest common ancestor in a binary search tree Number of visible boxes putting one inside another Program to calculate the area of the circumcircle of an equilateral triangle Red-black Tree in Data Structures Strictly binary tree in Data Structures 2-3 Trees and Basic Operations on them Asynchronous advantage actor-critic (A3C) Algorithm Bubble Sort vs Heap Sort Digital Search Tree in Data Structures Minimum Spanning Tree Permutation Sort or Bogo Sort Quick Sort vs Merge Sort Boruvkas algorithm Bubble Sort vs Quick Sort Common Operations on various Data Structures Detect and Remove Loop in a Linked List How to Start Learning DSA Print kth least significant bit number Why is Binary Heap Preferred over BST for Priority Queue Bin Packing Problem Binary Tree Inorder Traversal Burning binary tree Equal Sum What is a Threaded Binary Tree? What is a full Binary Tree? Bubble Sort vs Merge Sort B+ Tree Program in Q language Deletion Operation from A B Tree Deletion Operation of the binary search tree in C++ language Does Overloading Work with Inheritance Balanced Binary Tree Binary tree deletion Binary tree insertion Cocktail Sort Comb Sort FIFO approach Operations of B Tree in C++ Language Recaman’s Sequence Tim Sort Understanding Data Processing Applications of trees in data structures Binary Tree Implementation Using Arrays Convert a Binary Tree into a Binary Search Tree Create a binary search tree Horizontal and Vertical Scaling Invert binary tree LCA of binary tree Linked List Representation of Binary Tree Optimal binary search tree in DSA Serialize and Deserialize a Binary Tree Tree terminology in Data structures Vertical Order Traversal of Binary Tree What is a Height-Balanced Tree in Data Structure Convert binary tree to a doubly linked list Fundamental of Algorithms Introduction and Implementation of Bloom Filter Optimal binary search tree using dynamic programming Right side view of binary tree Symmetric binary tree Trim a binary search tree What is a Sparse Matrix in Data Structure What is a Tree in Terms of a Graph What is the Use of Segment Trees in Data Structure What Should We Learn First Trees or Graphs in Data Structures All About Minimum Cost Spanning Trees in Data Structure Convert Binary Tree into a Threaded Binary Tree Difference between Structured and Object-Oriented Analysis FLEX (Fast Lexical Analyzer Generator) Object-Oriented Analysis and Design Sum of Nodes in a Binary Tree What are the types of Trees in Data Structure What is a 2-3 Tree in Data Structure What is a Spanning Tree in Data Structure What is an AVL Tree in Data Structure Given a Binary Tree, Check if it's balanced B Tree in Data Structure Convert Sorted List to Binary Search Tree Flattening a Linked List Given a Perfect Binary Tree, Reverse Alternate Levels Left View of Binary Tree What are Forest Trees in Data Structure Compare Balanced Binary Tree and Complete Binary Tree Diameter of a Binary Tree Given a Binary Tree Check the Zig Zag Traversal Given a Binary Tree Print the Shortest Path Given a Binary Tree Return All Root To Leaf Paths Given a Binary Tree Swap Nodes at K Height Given a Binary Tree Find Its Minimum Depth Given a Binary Tree Print the Pre Order Traversal in Recursive Given a Generate all Structurally Unique Binary Search Trees Perfect Binary Tree Threaded Binary Trees Function to Create a Copy of Binary Search Tree Function to Delete a Leaf Node from a Binary Tree Function to Insert a Node in a Binary Search Tree Given Two Binary Trees, Check if it is Symmetric A Full Binary Tree with n Nodes Applications of Different Linked Lists in Data Structure B+ Tree in Data Structure Construction of B tree in Data Structure Difference between B-tree and Binary Tree Finding Rank in a Binary Search Tree Finding the Maximum Element in a Binary Tree Finding the Minimum and Maximum Value of a Binary Tree Finding the Sum of All Paths in a Binary Tree Time Complexity of Selection Sort in Data Structure How to get Better in Data Structures and Algorithms Binary Tree Leaf Nodes Classification of Data Structure Difference between Static and Dynamic Data Structure Find the Union and Intersection of the Binary Search Tree Find the Vertical Next in a Binary Tree Finding a Deadlock in a Binary Search Tree Finding all Node of k Distance in a Binary Tree Finding Diagonal Sum in a Binary Tree Finding Diagonal Traversal of The Binary Tree Finding In-Order Successor Binary Tree Finding the gcd of Each Sibling of the Binary Tree Greedy Algorithm in Data Structure How to Calculate Space Complexity in Data Structure How to find missing numbers in an Array Kth Ancestor Node of Binary Tree Minimum Depth Binary Tree Mirror Binary Tree in Data Structure Red-Black Tree Insertion Binary Tree to Mirror Image in Data Structure Calculating the Height of a Binary Search Tree in Data Structure Characteristics of Binary Tree in Data Structure Create a Complete Binary Tree from its Linked List Field in Tree Data Structure Find a Specified Element in a binary Search Tree Find Descendant in Tree Data Structure Find Siblings in a Binary Tree Given as an Array Find the Height of a Node in a Binary Tree Find the Second-Largest Element in a Binary Tree Find the Successor Predecessor of a Binary Search Tree Forest of a Tree in Data Structure In Order Traversal of Threaded Binary Tree Introduction to Huffman Coding Limitations of a Binary Search Tree Link State Routing Algorithm in Data Structure Map Reduce Algorithm for Binary Search Tree in Data Structure Non-Binary Tree in Data Structure Quadratic Probing Example in Hashing Scope and Lifetime of Variables in Data Structure Separate Chaining in Data Structure What is Dynamic Data Structure Separate Chaining vs Open Addressing Time and Space Complexity of Linear Data Structures Abstract Data Types in Data Structures Binary Tree to Single Linked List Count the Number of Nodes in the Binary Tree Count Total No. of Ancestors in a Binary Search Tree Elements of Dynamic Programming in Data Structures Find cost of tree with prims algorithm in data structures Find Preorder Successor in a Threaded Binary Tree Find Prime Nodes Sum Count in Non-Binary Tree Find the Right Sibling of a Binary Tree with Parent Pointers Find the Width of the Binary Search Tree Forest trees in Data Structures Free Tree in Data Structures Frequently asked questions in Tree Data Structures Infix, Postfix and Prefix Conversion Time Complexity of Fibonacci Series What is Weighted Graph in Data Structure Best Ways to Learn Data Structures and Algorithms Cardinality in ER Diagram Checking for the Mirror Images in the Binary Trees Complexity of Floyd Warshall Algorithm Count Non-Leaf Nodes in a Binary Tree Detect Cycle in Directed Graph using BFS Double Hashing in Data Structure Example of Dynamic Data Structure Expression Tree in Data Structure Floyd Warshall Algorithm Time Complexity Get a Parent Binary Tree Get All Descendants of the Binary Tree Element Get the Level of a Given Key in a Binary Tree Introduction to Data Structures Stack and Queue with Example Code in C++ Kadane Algorithm Max XOR of Two Numbers in an Array Mobile Numeric Keypad Problem Need for Data Structures and Algorithms for Deep Learning and Machine Learning The Knight's Tour Problem Traversal Techniques Weighted Graph in Data Structure What is Traversing in Data Structure Address Calculation Sort Alternative Sorting Applications of Linked List Load Factor and Rehashing in Data Structures Postfix Expression Using Stack Time Complexity of Job Sequencing with Deadlines 4 Queens Problem with the State Space Tree Dijkstra's Algorithm Augmented Data Structure Examples Binary Insertion Sort Difference Between Unary And Binary Operators Elements Of Dynamic Programming Lower Bound Theory Types of Data Structures in C Disjoint Set ADT in Data Structure What is an Algorithm Master Theorem Binary Tree to List in Order Binary Tree to Segment Tree Booth Algorithm Flowchart State Space Tree for 4 Queen Problem Primitive vs Non Primitive Data Structure Hash Function in Data Structure How to Implement Forward DNS Lookup Cache How to Implement Reverse DNS Lookup Cache Delete a Node from Linked List Find Loop in Linked List All-Pairs Shortest Paths Concurrency Algorithms Difference between Array and Structure Gcd Using Recursion Introduction to Backtracking Conversion of Binary to Gray Code

Perfect Binary Tree
2023-01-04 17:11:43

Perfect Binary Tree

Complete binary trees are an important and general topic in the concept – of tree data structures. Before discussing a complete binary tree, we need to know the concept of tree data structures.

Tree data structures

A non–linear data structure that is hierarchical is known as the tree data structure. Numerous data structures are used in computer science to organize data into different formats. A tree typically consists of a root value and child nodes that branch out of its parent nodes to form subtrees. Non-linear data structures include trees.

There is no cap on how many child nodes a standard tree data structure can store. The binary tree and its various varieties are a particular tree data structure that will be covered in this article.

 A tree data structure is classified into six types:

  1. Available trees.
  2. Binary trees.
  3. Binary search trees.
  4. AVL tress (Adelson, Velsky and Em Landis tree).
  5. Red – Black tree.
  6. N – ary Tree.

What is Binary Tree Data Structure?

A binary tree is a non-linear data structure in the shape of a tree that can have up to two children for each parent. In addition to the data element, each node in a binary tree also carries a left and right reference. A tree's root node is node at the top of the hierarchy. The parent nodes are the nodes that house the sub-nodes.

The left and right children are the two child nodes of a parent node. Some applications that use a binary tree include hashing, data compression, routing data for network traffic, creating binary heaps, and binary search trees.

Perfect binary tree

Terminologies associated with Binary Trees and Types of Binary Trees

  1. Node: In a tree, it serves as the branching point.
  2. Root: The topmost node of a tree.
  3. Parent: A parent node is any node in a tree (other than the root) with at least one sub-node.
  4. Child: When going away from the root, a child node is a node that immediately descends from a parent node.
  5. External nodes include leaf nodes. They are the nodes without offspring.
  6. Internal Node: These are inner nodes with at least one child, as the name implies.
  7. Depth: The depth of a tree is the number of edges that extend from its node to its root.
  8. Height: A tree's height is determined by counting the edges from the node to the deepest leaf.

Now that you know the terms used to describe binary trees and their many forms, it's time to comprehend the binary tree components.

Types of Binary Trees

There are different kinds of binary trees, and each kind has particular qualities. The specifics of each binary tree type are provided below:

  1. Full binary trees.
  2. Complete binary trees.
  3. Perfect binary trees.
  4. Balanced binary trees.
  5. Degenerate binary trees.

Perfect binary tree

If every exterior or leaf node is at the same level or depth inside a tree and all internal nodes have exactly two offspring, a binary tree is said to be "perfect." A height "h" perfect binary tree has two h - 1 node. A perfect binary tree looks like this:

Perfect binary tree Perfect binary tree

A sort of binary tree known as a perfect binary tree contains leaf nodes at the same level and inside nodes with exactly two children each.

Any internal node in a perfect binary tree has precisely two offspring, and all leaf nodes are located at the same level. This particular type of binary tree is known as a perfect binary tree.

The definition of a perfect binary tree is a binary tree in which each leaf node has the same level and either two offspring or none at all. The height or separation from the root node determines a node's level. A complete binary tree with a filled last level is called a perfect binary tree.

All internal nodes share a degree of 2.

If a single node has no children, it is a perfect binary tree with height h = 0. A node with h > 0 is a perfect binary tree if both subtrees are non-overlapping and have height h - 1.

Theorems of perfect binary tree

  1. A complete binary tree with height h has nodes that are two h + 1 - 1.
  2. The height of a perfect binary tree with n nodes is given by log(n + 1) - 1 = ln(n).
  3. There are two h leaf nodes in a perfect binary tree of height h.
  4. A node's average depth in a perfect binary tree is equal to ln(n).

Properties of perfect binary tree

  1. Degree: The degree of a node in a tree is based on how many children it has. All internal nodes have a degree of 2. The leaf nodes of a perfect binary tree have a degree of 0.
  2. Leaf node count: Since the last level is completely occupied, if the height of the perfect binary tree is h, the leaf node count will be two h.
  3. Node depth: The average depth of a node in a perfect binary tree is (ln(n)).
  4. Non-leaf nodes and leaf nodes are related. Non-leaf node count + one times the number of leaf nodes.
  5. Total nodes: A tree of height h has a total node count of 2h+1 - 1 node. The tree's nodes are all filled. The nodes' sum is 20 + 21 +... + 2h = 2h+1 - 1.
  6. Tree height: A perfect binary tree with N nodes has a height of log(N + 1) - 1 = ln(n). When computing the total number of nodes in a perfect binary tree, the relationship indicated can be used to compute this.

To check whether a given tree is a perfect binary tree or not:

  1. Determine any node's depth (in the below tree, we find the depth of the leftmost node). Let d be the depth.
  2. Recursively go through the tree now and look for the following two circumstances.
    1. Each internal node must have two non-empty children.
    1. Every leaf is at a depth.