# AA Trees | Set 1 (Introduction)

AA trees are the variation of the red-black trees, a form of binary search tree.

AA trees use the concept of **levels** to aid in **balancing binary trees**. The **level** of node (instead of colour) is used for balancing information. A link where child and parent’s levels are same, is called a horizontal link, and is analogous to a red link in the red-black tree.

- The level of every leaf node is one.
- The level of red nodes are same as the level of their parent nodes and the links are called
**horizontal links**. - The level of black nodes are one less than the level of their parent node.

Additional storage requirement with every node is O(Log n) in red black trees instead of O(1) (only color in Red Black Trees), but AA trees simplify restructuring by removing many cases.

An AA tree follows same rule as red-black trees with the addition of single new rule that red nodes cannot be present as left child.

- Every node can be either red (linked horizontally) or black.
- There are no two adjacent red nodes (or horizontal links).
- Every path from root to a NULL node has same number of black nodes (ot black links).
**Left link cannot NOT be red (horizontal).***(New added rule)*

**Why AA trees :**

The implementation and number of rotation cases in Red-Black Trees is complex. AA trees simplifies the algorithm.

- It eliminates half of the restructuring process by eliminating half of the rotation cases, which is easier to code.
- It simplifies the deletion process by removing multiple cases.

Below tree is the example of AA tree :

Note that in the above tree there are no left red child which is the new added rule of AA Trees.

After re-drawing the above AA tree with levels and horizontal links (the red nodes are shown connected through horizontal or red links), the tree looks like:

Note that all the nodes on level 1 i.e. 5, 10, 20, 35, 40, 55, 65, 80, 90 are known as leaf nodes.

**So, in summarized way, for tree to be AA tree, it must satisfy the following five invariants:**

- Wavelet Trees | Introduction
- proto van Emde Boas Trees | Set 1 (Background and Introduction)
- Generic Trees(N-array Trees)
- B*-Trees implementation in C++
- m-WAY Search Trees | Set-1 ( Searching )
- Check if two trees are Mirror | Set 2
- Combinatorics on ordered trees
- Dynamic Programming on Trees | Set-1
- Dynamic Programming on Trees | Set 2
- Disjoint Set Union on trees | Set 1
- Disjoint Set Union on trees | Set 2
- Check if two trees have same structure
- Check if two trees are Mirror
- Red-Black Trees | Top-Down Insertion
- Some Basic Theorems on Trees

*1.) The level of leaf node is 1.*

2.) The level of left child is exactly one less than of its parent.

3.) The level of every right child is equal to or one less than of its parent.

4.) The level of every right grandchild is strictly less than that of its grandparent.

5.) Every node of level greater than one has two children.

2.) The level of left child is exactly one less than of its parent.

3.) The level of every right child is equal to or one less than of its parent.

4.) The level of every right grandchild is strictly less than that of its grandparent.

5.) Every node of level greater than one has two children.

**References :**

http://www.eternallyconfuzzled.com/tuts/datastructures/jsw_tut_andersson.aspx

https://ycpcs.github.io/cs350-fall2017/lectures/AA-tree_lecture.pdf

https://en.wikipedia.org/wiki/AA_tree

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