1. Red Black Trees

2. So…
Btrees self balance
Can represent as a binary tree…

3. Red Black Tree Red Black Tree = binary version of BTree
Red nodes are part of their parent
1 red + 1 black = node degree 3
2 red + black = node degree 4

4. Red Black vs BTree Two views of same tree
Btree degree 4: Red Black Tree

5. Red Black Rules
Rules
The root is black

6. Red Black Rules
Rules
The root is black – if it becomes red, turn it back to black
Null values are black
A red node must have black children
Every path from root to leaf must have same number of black nodes

7. Guarantee
Worst and best case in terms of red nodes for black height = 2
If L = num leaves, B = black height B ≤ L ≤ 22B

8. Guarantee So… 2 𝐵 ≤ 𝐿 ≤ 2 2𝐵 𝐵 ≤ 𝑙𝑜𝑔 2 𝐿 ≤2𝐵
1 𝐵 ≥ 1 𝑙𝑜𝑔 2 𝐿 ≥ 1 2𝐵 𝑙𝑜𝑔 2 𝐿 𝐵 ≥ ≥ 𝑙𝑜𝑔 2 𝐿 2𝐵 𝑙𝑜𝑔 2 𝐿 ≥ 𝐵 ≥ 𝑙𝑜𝑔 2 𝐿 2

9. Height 𝑙𝑜𝑔 2 𝐿 ≥ 𝐵 Black height is O(logL) Total height at most 2B
2O(logL) = O(logL)
Height is O(logL)
Total nodes (N) < 2L – 1
O(log(n/2)) = O(logn)  Guaranteed logN performance
𝑙𝑜𝑔 2 𝐿 ≥ 𝐵

10. Actual Work Insert as normal New node is always red
Two red's in a row need to be fixed…

11. Fixes Red child, red parent, no uncle or black uncle
Zig-zag style double rotation
New parent becomes black, new child red
== Rearranging values in 4 node of BTree

12. Fixes Red child has red parent and red uncle
Push up redness of siblings to grandparent
Fix at grandparent if new problem
If root becomes red, make it black
== Splitting a node with 4 keys in BTree

13. RB Deletions Need to preserve no red->red
Need to preserve consistent black height
Tools : Recolors and rotates

14. RB Deletions
Lots of special cases… some easy, some tricky

15. Binary Tree Comparisons
Plain BST
Maintains data in sorted order
Hopefully O(logn)
Could be O(n)

16. Binary Tree Comparisons
Splay
Pull nodes to
Amortized O(logn)
Ideal for consecutive accesses

17. Binary Tree Comparisons
AVL
Guaranteed O(logn)
Height limited to ~1.44 log2(n)
High constant factors on insert/delete

18. Binary Tree Comparisons
Red/Black
Guaranteed O(logn)
Height limited to ~2 log2(n)
Less balanced than AVL
Faster insert/remove
Slower find
Standard implementation for most library BSTs

19. Tree Comparisons BTree Not binary – can pick any node size
Still sorted
Self balancing – log(n) performance
Ideal for slower storage
Model for red/black trees

20. Tree Comparisons Other trees Represent tree structure
Not necessarily sorted