Binary Search Tree

Lesson notes

Overview

Smaller keys go left, larger go right, so lookups skip half the tree at each step. This topic is rated beginner and is number 21 of 117 in Computer Science. Core ideas and first intuitions. No prior knowledge needed.

What you are seeing

The panel above shows an interactive animation of the idea. Every slider changes one quantity so you can see its effect on the outcome. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

A binary search tree stores keys so that everything in a node’s left subtree is smaller and everything in its right subtree is larger. Searching, inserting and deleting follow one path down the tree, skipping half of the remaining items at each step, which takes O(log n) when the tree is balanced. If data is inserted in sorted order the tree degenerates into a list and takes O(n), so self-balancing variants such as AVL and red-black trees are used in practice.

Reading the relationship

The expression "O(log n) if balanced" is a compact summary of the idea. It uses the symbols O, n, i, f, b, a; name what each one stands for before you start. Notice that it contains a logarithm, which turns multiplication into addition and compresses large values. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Run the animation once with the default settings and describe in words what happens. 2. Change one slider at a time and note which quantity responds most. 3. Push each slider to its extreme and decide whether the result still makes sense.

Try this

Start by only moving one slider at a time. Say aloud what you expect, then check.

Common pitfalls

Test small inputs by hand and then the largest ones. Off-by-one errors and forgetting how cost grows with input size are the most common traps.

How it connects

It builds on “Queue”, the topic before it in Computer Science. Once you are comfortable, continue with “Binary Heap”.

Why it matters

Computing ideas decide how fast, how safely and how well software can solve problems.