AA trees use the concept of levels to aid in balancing binary trees. The level of node (instead of colour) is used as as 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:
1.) The level of leaf node is 1.
- Euler Tour | Subtree Sum using Segment Tree
- Range and Update Query for Chessboard Pieces
- XOR of numbers that appeared even number of times in given Range
- Number of primes in a subarray (with updates)
- Levelwise Alternating OR and XOR operations in Segment Tree
- Range query for Largest Sum Contiguous Subarray
- Dynamic Disjoint Set Data Structure for large range values
- Leftover element after performing alternate Bitwise OR and Bitwise XOR operations on adjacent pairs
- Range Queries for Longest Correct Bracket Subsequence
- Comparisons involved in Modified Quicksort Using Merge Sort Tree
2.) The level of left child is exactly one less than of its parent.
3.) The level of every right child is 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.
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