11 Sep - DS - Trees Part - 1
Duration: 1 hr 5 min
This video lesson is available to enrolled students.
AI Summary
An AI-generated summary of this video lecture.
This educational video provides a comprehensive lecture on Tree data structures, beginning with fundamental definitions and progressing to specific problem-solving techniques for competitive exams like GATE. The instructor starts by defining a tree as a nonlinear data structure representing hierarchical relationships, highlighting key terms such as root node, parent-child relationships, and subtrees using a detailed diagram with nodes labeled A through K. The lecture then transitions to defining specific tree terminology including edges, leaf nodes, internal nodes, degree, and level/depth. A significant portion is dedicated to Binary Trees, covering their properties, memory representations (sequential and linked), and the C struct implementation. The latter half of the video focuses on solving four distinct GATE exam questions (2017, 2015, 2010, and 2005) that test concepts of tree height, node counts, and traversal properties, with the instructor demonstrating step-by-step derivations and logical reasoning for each problem.
Chapters
0:00 – 2:00 00:00-02:00
The video begins with an introductory slide titled 'Tree'. The text defines a tree as a nonlinear data structure that represents hierarchical relationships between data items. It emphasizes the flexibility and versatility of trees. The slide introduces the concept of the root node as the starting point and explains that each node, except the root, is connected to exactly one parent node, forming a parent-child relationship. The instructor is visible in the top right corner, introducing the topic.
2:00 – 5:00 02:00-05:00
The slide continues with bullet points defining specific tree components. 'Root Node' is described as the top-most node in the tree. 'Subtrees' are defined as smaller subsets of the tree, each acting as an independent tree. 'Applications' are listed, including file systems, database indexing, and decision-making. Below the text, a large diagram of a tree is displayed with nodes labeled A through K in blue circles, connected by lines representing edges. Node A is at the top as the root.
5:00 – 10:00 05:00-10:00
The instructor explains the tree diagram shown on the slide. He points out that node A is the root node. He traces the connections to show that A is the parent of B and C. He explains that the tree can be divided into subtrees rooted at B and C. He highlights the hierarchical nature, showing how B is a parent to D, E, and F, and C is a parent to G and H. He further details that E is a parent to I and J, and G is a parent to K.
10:00 – 15:00 10:00-15:00
A new slide appears listing detailed definitions of tree terminology. The terms include Root Node, Edge, Child Node, Parent Node, Leaf (External) Node, Internal Node, Degree of Node, Level/Depth, Path, and Subtree. The instructor begins to go through these definitions, explaining that an edge is the connection between two nodes. He notes that a tree with 'N' nodes has exactly 'N-1' edges. He defines a child node as one that descends from a parent node.
15:00 – 20:00 15:00-20:00
The instructor continues explaining the definitions on the slide. He highlights 'Leaf (External) Node' as a node with no children and 'Internal Node' as a node with at least one child. He explains 'Degree of Node' as the number of children a node has. He defines 'Level/Depth' as the step count from the root (Level 0) to a particular node. He defines a 'Path' as a sequence of nodes connected by edges and reiterates the definition of a 'Subtree' as a tree formed from a child node and its descendants.
20:00 – 25:00 20:00-25:00
The slide changes to 'Binary tree'. It defines a binary tree as a type of tree data structure where each node can have at most two children, commonly referred to as left and right child nodes. The slide lists properties: 'Empty Tree' (no nodes), 'Root Node' (unique if not empty), and 'Subtrees' (partitioned into T1 left and T2 right). It also discusses 'Representation of tree in memory', mentioning Sequential representation (array) and Linked Representation (self-referential structure node). A C struct definition is shown with fields `int data`, `struct node* left`, and `struct node* right`.
25:00 – 30:00 25:00-30:00
The instructor elaborates on the binary tree slide. He draws a simple binary tree structure on the screen to illustrate the concept of left and right children. He explains that in a binary tree, a node can have zero, one, or two children. He points to the C struct code, explaining that `struct node* left` and `struct node* right` are pointers to the left and right child nodes, respectively. He emphasizes that this self-referential structure allows for the creation of linked binary trees in memory.
30:00 – 35:00 30:00-35:00
A GATE 2017 question appears on the screen: 'Let T be a binary search tree with 15 nodes. The minimum maximum possible heights of T are: ______'. The options are (A) 4 and 15, (B) 3 and 14, (C) 4 and 14, (D) 3 and 15. The instructor begins to solve the problem by drawing a complete binary tree to find the minimum height. He writes down the formula for the maximum number of nodes in a binary tree of height h, which is $2^{h+1} - 1$.
35:00 – 40:00 35:00-40:00
The instructor continues solving the GATE 2017 question. He calculates the maximum height for 15 nodes, which occurs in a skewed tree where each node has only one child. In this case, the height is $n-1$, so $15-1 = 14$. He then calculates the minimum height by filling the tree level by level. He writes $2^0 + 2^1 + 2^2 + 2^3 = 1 + 2 + 4 + 8 = 15$. This means a complete binary tree with height 3 can hold exactly 15 nodes.
40:00 – 45:00 40:00-45:00
The instructor concludes the solution for the GATE 2017 question. He confirms that the minimum height is 3 and the maximum height is 14. He selects option (B) 3 and 14 respectively. He circles the correct option on the slide. He emphasizes that for a binary search tree, the minimum height is achieved when the tree is balanced (complete), and the maximum height is achieved when the tree is skewed.
45:00 – 50:00 45:00-50:00
A new GATE 2015 question appears: 'The height of a tree is the length of the longest root-to-leaf path. The maximum and minimum number of nodes in a binary tree of height 5 are ______'. The options are (A) 63 and 6, (B) 64 and 5, (C) 32 and 6, (D) 31 and 5. The instructor starts solving this by calculating the maximum number of nodes. He uses the formula $2^{h+1} - 1$. For height 5, this is $2^{5+1} - 1 = 2^6 - 1 = 64 - 1 = 63$.
50:00 – 55:00 50:00-55:00
The instructor continues with the GATE 2015 question. He calculates the minimum number of nodes for a binary tree of height 5. He explains that the minimum number of nodes occurs when the tree is a straight line (skewed), so the number of nodes is $h + 1$. For height 5, this is $5 + 1 = 6$. He writes 'max -> 63' and 'min -> 6' on the screen. He identifies option (A) 63 and 6 respectively as the correct answer.
55:00 – 60:00 55:00-60:00
A GATE 2010 question is displayed: 'In a binary tree with n nodes, every node has an odd number of descendants. Every node is considered to be its own descendant. What is the number of nodes in the tree that have exactly one child?'. The options are (A) 0, (B) 1, (C) (n-1)/2, (D) n-1. The instructor analyzes the condition 'odd number of descendants'. He draws a node with one child and counts the descendants: the child itself plus the node itself equals 2, which is even. This contradicts the condition.
60:00 – 65:00 60:00-65:00
The instructor continues analyzing the GATE 2010 question. He draws a node with two children. The descendants are the two children plus the node itself, totaling 3, which is odd. This satisfies the condition. He concludes that for every node to have an odd number of descendants, every node must have either 0 children (1 descendant: itself) or 2 children (3 descendants: itself + 2 children). Therefore, no node can have exactly one child. The answer is 0, which corresponds to option (A). He circles option (A).
65:00 – 65:04 65:00-65:04
The video concludes with the instructor finishing the explanation of the GATE 2010 question. The screen shows the final answer circled. The instructor is visible in the top right corner, wrapping up the lecture segment.
The lecture systematically builds knowledge from basic tree definitions to complex problem-solving. It starts by establishing the vocabulary of tree data structures (root, parent, child, leaf, internal node) using a visual diagram. It then narrows the focus to binary trees, explaining their specific constraints (max 2 children) and memory representation via C structs. The core of the lecture involves applying these theoretical concepts to solve past GATE exam questions. The instructor demonstrates how to calculate minimum and maximum heights and node counts using formulas like $2^{h+1}-1$ and $h+1$. He also uses logical deduction to solve a problem about descendant counts, proving that a specific condition implies a binary tree must be a full binary tree (0 or 2 children). This progression from definition to application provides a robust review for students preparing for data structure exams.