BST Practice Question

Duration: 13 min

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Module outline

  1. About the course: About the Course
  2. Mathematical Aptitude: Number System, HCF LCM, Divisibility Rules, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Time and Work, Mensuration and Geometry, Simple Interest and Compound Interest, Speed Time and Distance, Age Problems, Mixture and Alligation, Algebra, Powers and Exponents (Surds and Indices), Pipe and Cistern, Trigonometry, Statistics, Data Interpretation, Arithmetic
  3. Logical Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Analogy, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Seating Arrangements, Syllogisms, Venn Diagram, Puzzle Solving (Puzzles), Logical Sequence Of Words, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Analytical Reasoning (Counting Figures Reasoning), Cubes & Dices, Calendar, Clock
  4. General Knowledge: Physics, Chemistry, Biology, History, Geography, Polity, Indian Economy, GK- Current Affairs, Static General Knowledge
  5. Art of Teaching: Nature, Objectives & Characteristics of Teaching, Learners & Learning Process, Factors Affecting Teaching, Methods of Teaching, Classroom Management & Communication, Teaching-Learning Aids & ICT Integration, Evaluation, Assessment & Measurement, Professional Development & Ethics, Policies, Acts & Contemporary Issues, Curriculum & School Environment, Guidance, Counselling & Mental Health, Inclusive Education & Special Needs, Micro-Teaching & Teaching Skills, Lesson Planning
  6. Environmental Science: Environment
  7. Unit 1 | Digital Electronics: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation
  8. Unit 2 | COA: Basics of COA, Floating Point Rep, Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Control Unit Design, Register Transfer and Microoperations, Programming the Basic Computer
  9. Unit 3 | C Programming: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling
  10. Unit 3 | Data Structures: Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  11. Unit 4 | Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  12. Unit 5 | Operating System: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management
  13. Unit 6 | Database Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Transaction Management, Concurrency Control
  14. Unit 7 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer
  15. Unit 8 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  16. Unit 9 | Object Oriented Programming: OOPS with C++
  17. Unit 10 | Web Development: HTML Basics, CSS, JavaScript-Basics, XML
  18. Unit 11 | Theory Of Computation: Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Grammar, Regular Language Properties, Moore & Mealy Machines, Pushdown Automata & CFG, Turing Machines
  19. Unit 12 | IoT: Emerging Tech Trends, Cloud Computing
  20. Unit 13 | Artificial Intelligence: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Natural Language Processing, Artificial Neural Networks, Genetic Algorithms
  21. Unit 14 | E-Commerce: Electronic Data Interchange and E-Commerce, Role of Internet and Web Services, Security in Mobile and Wireless Computing, Physical Security and Cryptography Basics, Cyber Safety and Security
  22. Unit 15 | Multimedia: Multimedia
  23. Unit 16 | Python: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis, DSA using Python
  24. Full Length Mock Tests — General Paper:
  25. Full Length Mock Tests — Computer Science (Paper II):
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This lecture solves a binary search tree (BST) counting problem by introducing Catalan numbers. It begins with the example of 3 distinct keys, showing that there are 5 possible BST structures. The main question asks how many distinct BSTs can be created from 4 distinct keys, with options (A) 4, (B) 14, (C) 24, and (D) 42. The instructor introduces the Catalan number formula in both recursive form, C_n = sum_{i=0}^{n-1} C_i * C_{n-i-1}, and closed form, (2n choose n)/(n+1) or (2n)!/((n+1)! * n!). A table is built for f(N) where N represents the number of keys: f(0)=1, f(1)=1, f(2)=2, and f(3)=5. For N=4, the instructor considers each key (1, 2, 3, or 4) as the root and multiplies the number of left-subtree combinations by right-subtree combinations: C_0*C_3 + C_1*C_2 + C_2*C_1 + C_3*C_0 = 5 + 2 + 2 + 5 = 14. Option (B) 14 is circled as the correct answer.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a practice question asking how many distinct BSTs can be constructed with 3 distinct keys, displaying the 5 possible tree structures for N=3. It then transitions to the main question: 'How many distinct binary search trees can be created out of 4 distinct keys?' with options (A) 4, (B) 14, (C) 24, and (D) 42. The instructor writes 'Formula of catalan number' in red, showing the recurrence c_n = sum_{i=0}^{n-1} c_i * c_{n-i-1} and the closed form (2n choose n)/(n+1). A two-column table headed 'N' and 'f(n)' is drawn, with the row N=0 entered as f(0)=1.

  2. 2:00 – 5:00 02:00-05:00

    The instructor builds the table step by step, writing N=1 with f(1)=1=C_1 and continuing to calculate values for increasing N. The closed-form formula C_n = (2n)!/((n+1)! * n!) is displayed on the right side of the screen. For N=3, the instructor draws explicit BST structures using keys 1, 2, and 3 to visualize why f(3)=5. The recursive formula C_n = sum_{i=0}^{n-1} C_i * C_{n-i-1} is written alongside the table, connecting the manual enumeration to the general Catalan number recurrence.

  3. 5:00 – 10:00 05:00-10:00

    The instructor calculates f(2)=2 and confirms f(3)=5 by summing products of left and right subtree combinations. For N=4, the instructor lists all possible roots (1, 2, 3, and 4) and draws tree diagrams for each root choice. The breakdown shows that choosing key 1 as root gives C_0*C_3 combinations, key 2 gives C_1*C_2, key 3 gives C_2*C_1, and key 4 gives C_3*C_0. The table now shows f(0)=1, f(1)=1, f(2)=2, and f(3)=5, with the Catalan recurrence C_n = sum C_i * C_{n-1-i} displayed on the right.

  4. 10:00 – 12:58 10:00-12:58

    The final calculation sums the products for N=4: C_0*C_3 + C_1*C_2 + C_2*C_1 + C_3*C_0 = 5 + 2 + 2 + 5 = 14. The instructor writes f(4)=14 in the table and circles option (B) 14 as the correct answer. The completed whiteboard shows the full table from N=0 to N=4, four BST structures drawn at the bottom corresponding to roots 1 through 4, and the Catalan number formula C_n = sum_{i=0}^{n-1} C_i * C_{n-i-1} on the right side, verifying that the recursive counting matches the closed-form Catalan number.

The central concept is that the number of distinct BSTs with n keys equals the nth Catalan number. The teaching flow moves from a concrete example (N=3, five trees) to the general formula. The recursive method works by choosing each key as root: if key i is root, the left subtree has (i-1) keys and the right subtree has (n-i) keys, contributing C_{i-1} * C_{n-i} combinations. Summing over all roots gives the recurrence. The closed form (2n)!/((n+1)! * n!) provides a direct calculation. For N=4, the answer 14 is verified both by the recurrence and by explicit enumeration of root cases. Key exam takeaway: recognize BST counting problems as Catalan number applications, and be able to compute small values (1, 1, 2, 5, 14) from the recurrence.

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