Project 9: Search Tree Collection (BST, AVL, B-Trees)

Implement and compare core search trees from TAOCP Vol 3.

Quick Reference

Attribute Value
Difficulty Expert
Time Estimate 3-4 weeks
Language C
Prerequisites Trees, rotations, invariants
Key Topics balancing, height bounds, performance

1. Learning Objectives

By completing this project, you will:

  1. Implement BST insert/search/delete.
  2. Implement AVL rotations and rebalancing.
  3. Implement B-tree split/merge operations.
  4. Validate invariants and compare performance.

2. Theoretical Foundation

2.1 Core Concepts

  • BST invariant: left < node < right.
  • AVL balance factor: height difference <= 1.
  • B-tree order: node capacity constraints and balanced height.

2.2 Why This Matters

Search trees are the backbone of indexing and efficient lookup in databases and filesystems.

2.3 Historical Context / Background

Knuth’s Vol 3 formalizes search tree analysis and practical balancing rules.

2.4 Common Misconceptions

  • “AVL is optional”: Without balance, height degrades.
  • “B-trees are just big BSTs”: They optimize for block storage.

3. Project Specification

3.1 What You Will Build

A library implementing BST, AVL, and B-tree with insert/search/delete and validation tooling.

3.2 Functional Requirements

  1. BST with search/insert/delete.
  2. AVL with rotations and height tracking.
  3. B-tree with split/merge and search.
  4. Benchmarks for height and operations.

3.3 Non-Functional Requirements

  • Correctness: invariants always hold.
  • Performance: height within theoretical bounds.
  • Usability: common API and tests.

3.4 Example Usage / Output

insert 10,20,30 -> AVL rotates
height=2

3.5 Real World Outcome

You can demonstrate why balanced trees outperform naive BSTs on sorted input.

4. Solution Architecture

4.1 High-Level Design

┌──────────────┐     ┌──────────────┐     ┌──────────────┐
│ tree API     │────▶│ bst/avl/btree│────▶│ benchmarks   │
└──────────────┘     └──────────────┘     └──────────────┘

4.2 Key Components

Component Responsibility Key Decisions
BST core Basic operations recursion vs iteration
AVL core Balance maintenance rotation helpers
B-tree core multiway nodes order selection

4.3 Data Structures

typedef struct BstNode {
    int key;
    struct BstNode *left;
    struct BstNode *right;
} BstNode;

4.4 Algorithm Overview

Key Algorithm: AVL insert

  1. Insert like BST.
  2. Update heights.
  3. Rotate if balance factor out of range.

Complexity Analysis:

  • Time: O(log n) for AVL, O(log n) for B-tree.
  • Space: O(n).

5. Implementation Guide

5.1 Development Environment Setup

gcc --version

5.2 Project Structure

project-root/
├── include/trees.h
├── src/bst.c
├── src/avl.c
├── src/btree.c
├── tests/
└── Makefile

5.3 The Core Question You’re Answering

“How do we guarantee logarithmic search time with invariants?”

5.4 Concepts You Must Understand First

  • Rotations and height updates
  • B-tree node capacity constraints
  • Validation of ordering invariants

5.5 Questions to Guide Your Design

  1. How will you validate balance factors?
  2. What B-tree order is appropriate for tests?
  3. How will you handle delete cases?

5.6 Thinking Exercise

Prove why AVL height is O(log n).

5.7 The Interview Questions They’ll Ask

  1. Why does a BST degrade to O(n)?
  2. What does a rotation preserve?
  3. Why are B-trees ideal for disks?

5.8 Hints in Layers

Hint 1: Implement BST first. Hint 2: Add AVL rotations. Hint 3: Add B-tree splitting.

5.9 Books That Will Help

Topic Book Chapter
Search trees TAOCP Vol 3 Ch. 6.2

5.10 Implementation Phases

Phase 1: Foundation (1 week)

  • BST with validation and tests.

Phase 2: Core Functionality (1-2 weeks)

  • AVL rotations and balancing.

Phase 3: Polish & Edge Cases (1 week)

  • B-tree split/merge and benchmarks.

5.11 Key Implementation Decisions

Decision Options Recommendation Rationale
AVL storage height vs balance height simpler update
B-tree order 3, 4, 8 4 manageable test size

6. Testing Strategy

6.1 Test Categories

Category Purpose Examples
Invariants correctness BST order checks
Balancing AVL rotations sorted inserts
B-tree split/merge node capacity tests

6.2 Critical Test Cases

  1. Insert sorted input into AVL stays balanced.
  2. Delete nodes in AVL rebalances correctly.
  3. B-tree split maintains order.

6.3 Test Data

Sorted, random, reverse sequences

7. Common Pitfalls & Debugging

7.1 Frequent Mistakes

Pitfall Symptom Solution
Missing rotation height grows check balance on insert/delete
Wrong pointer updates crashes test small cases
B-tree underflow invalid nodes implement merge/borrow

7.2 Debugging Strategies

  • Add invariant checks after each op.
  • Print tree structure for small tests.

7.3 Performance Traps

Validation on every operation can be slow; toggle with debug flag.

8. Extensions & Challenges

8.1 Beginner Extensions

  • Add red-black tree.

8.2 Intermediate Extensions

  • Add order-statistics tree.

8.3 Advanced Extensions

  • Implement B+ tree variant.

9. Real-World Connections

9.1 Industry Applications

  • Database indices and filesystem metadata.
  • sqlite and btree implementations.

9.3 Interview Relevance

  • Balanced trees are a common interview topic.

10. Resources

10.1 Essential Reading

  • TAOCP Vol 3 Chapter 6.2.

10.2 Video Resources

  • Search: “AVL rotations”.

10.3 Tools & Documentation

  • None beyond standard C tooling.

11. Self-Assessment Checklist

11.1 Understanding

  • I can explain AVL rotations.
  • I can describe B-tree node constraints.

11.2 Implementation

  • All invariants pass tests.
  • Benchmarks show expected heights.

11.3 Growth

  • I can choose an appropriate tree for a workload.

12. Submission / Completion Criteria

Minimum: BST and AVL correct. Full: B-tree with tests. Excellence: Additional balanced tree variant.

This guide was generated from LEARN_TAOCP_DEEP_DIVE.md. For the complete learning path, see the parent directory README.