1. 1 /26 Red-black tree properties Every node in a red-black tree is either black or red Every null leaf is black No path from a leaf to a root can have two consecutive red nodes -- i.e. the children of a red node must be black Every path from a node, x, to a descendant leaf contains the same number of black nodes -- the “black height” of node x.

2. 2 /26 Red-black tree insertion: Case 0: New node’s parent is black. There’s nothing to do. newly inserted node

3. 3 /26 Red-black tree insertion: Case 1: New node’s parent and uncle are both red. color flip: x’s parent, uncle, and grandparent all change color newly inserted node this edge/node is now red; possibly propagating a r-b violation upward x = (x could also be a right child) compare w/ Figure 14.5 x =

4. 4 /26 Red-black tree insertion: Case 3: New red node’s parent is red and uncle is black ; x is a left child & p(x) is a left child Flip color of p(x) and p(p(x)), then do a right rotation at p(p(x)) newly red node (caused by propagation from below)     AB C A BC x = x See Fig. 14.6

5. 5 /26 Red-black tree insertion: Case 2: New red node’s parent is red and uncle is black; x is a right child & p(x) is a left child Double rotation: rotate once at p(x), rotate again at the new p(x) newly red node (caused by propagation from below)    x A B D A CD C x  B

6. 6 /26 Red-black tree insertion: Case 2a: New red node’s parent is red and uncle is black; x is a right child & p(x) is a left child x = p(x); Rotate left at new x =  Applying the code in the box above reduces case 2a to case 3. newly red node (caused by propagation from below)   A B D C x =   AB D C

7. 7 /26 Red-black tree insertion: Case 2b: New red node’s parent is red and uncle is black; x is a right child & p(x) is a left child x = p(x); Rotate left at new x Case 2b: Same as Case 3 described earlier. newly red node (caused by propagation from below)   A B D C x =  A  BDC

8. 8 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 

9. 9 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6   Insert 9 and flip color of root

10. 10 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6      Need to apply case 2: rotate at 9, then rotate at 2.

11. 11 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6      Need to apply case 2: rotate at 9 …

12. 12 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6      Need to apply case 2: … then rotate at 2

13. 13 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6     Red-black property restored!  1

14. 14 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6   Red-black property violated by inserting 8. An application of case 1 suffices here     8

15. 15 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6   A color flip restores the red-black property.  2    8

16. 16 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 Inserting 3 violates the red-black property    8    8 3

17. 17 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 This is case 3: Rotate right at 9 to restore.    8    8 3

18. 18 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 This is case 3: Rotate right at 9 and flip colors to restore.    8    3 9

19. 19 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 Inserting 7 causes a red-black violation; both 7’s parent and uncle are red, so a color flip suffices …    3 9    3 9 7

20. 20 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 The color flip performed: 3, 9, and 8 all flip.         3 9 7

21. 21 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6 Inserting 4 causes a red-black violation; perform a right rotation at 7... then a left rotation at (the new position of) 4...      7       4 

22. 22 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6      7       7 

23. 23 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6      7       7   Inserting 6 causes a red-black violation … which can be fixed by a color flip (it’s a case 1 violation)

24. 24 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6         But flipping the colors of 6’s parent, uncle, and grandparent causes another tree violation at 4...      7  

25. 25 /26 Example: Inserting the sequence 1 9 2 8 3 7 4 6        This situation is covered by case 2 and results in a double rotation (two single rotations shown here in combination as one double rotation).        

26. 26 /26 Inserting 1 9 2 8 3 7 4 6 into a binary search tree without balancing! 2 8 1     3 6