---
id: avl
title: AVL tree
description: |
  Looking at a tree that's very similar to the red-black tree.
tags:
  - AVL tree
  - balanced trees
  - zig
last_update:
  date: 2024-01-22
---

## Introduction

AVL tree is the first self-balanced tree to be ever invented. However if we look
at the commonly used data structures to represent the set or a map, we will find
out that red-black trees (or even its relaxation LLRB[^1]) or hash tables are
the most used.

Let's look into the AVL tree and how it is related to its “successor” red-black
tree, they are closer than you can imagine.

## Properties

AVL tree is height-balanced same as the red-black tree. The key difference lies
in the invariants that need to be satisfied. In this case we require that
subtrees of any node differ in their heights at most by 1. If we were to write
it as a condition for node $n$, we can write it like:

$$
\vert Height(Right(n)) - Height(Left(n)) \vert \leq 1
$$

Because the only allowed values here are $\left\{ -1, 0, 1 \right\}$, it is
possible to simplify the structure of the nodes in memory by using a one byte of
memory to store the “state” of the subtrees in a node.

:::tip

Some representations use a _sign_ notation of `+`, `0` and `-` which represent
where is the tree leaning to:

- `-` means the left subtree has bigger height ⇒ tree is leaning to left
- `0` means both subtrees have the same height ⇒ they're equal
- `+` means the right subtree has bigger height ⇒ tree is leaning to right

:::

It is also possible to find representations that allow leaning only to one side,
by enforcing this requirement you can use just boolean to represent the state,
i.e. the tree _is_ leaning to one side, or _is not_.

## Implementation

Let's have a look at the implementation of the AVL tree in some language.

## Insertion

## Deletion

## Comparison with the red-black tree

## Bonus: Implementation of the BST in Java

[^1]: [Left-leaning Red-Black trees](https://sedgewick.io/wp-content/themes/sedgewick/papers/2008LLRB.pdf)