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.

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

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 /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 /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 /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 /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 /26 Example: Inserting the sequence 

9 /26 Example: Inserting the sequence   Insert 9 and flip color of root

10 /26 Example: Inserting the sequence      Need to apply case 2: rotate at 9, then rotate at 2.

11 /26 Example: Inserting the sequence      Need to apply case 2: rotate at 9 …

12 /26 Example: Inserting the sequence      Need to apply case 2: … then rotate at 2

13 /26 Example: Inserting the sequence     Red-black property restored!  1

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

15 /26 Example: Inserting the sequence   A color flip restores the red-black property.  2    8

16 /26 Example: Inserting the sequence Inserting 3 violates the red-black property    8    8 3

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

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

19 /26 Example: Inserting the sequence 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 /26 Example: Inserting the sequence The color flip performed: 3, 9, and 8 all flip.         3 9 7

21 /26 Example: Inserting the sequence 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 /26 Example: Inserting the sequence      7       7 

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

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

25 /26 Example: Inserting the sequence        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 Inserting into a binary search tree without balancing!     3 6