1. Red-Black Trees by Thomas A. Anastasio

2. A Red-Black Tree with NULLs shown Black-Height of the tree = 4

4. A valid Red-Black Tree Black-Height = 2

13. X P G U P G U Case 1 – U is Red Just Recolor and move up X

14. X P G U S X P G S U Case 2 – Zig-Zag Double Rotate X around P; X around G Recolor G and X

15. X P G U S P X G S U Case 3 – Zig-Zig Single Rotate P around G Recolor P and G

16. 11 14 15 2 1 7 5 8 Black node Red node Insert 4 into this R-B Tree

19. Red-Black Trees Bottom-Up Deletion

26. Case 1 Diagram P S V+ P S V+ Rotate P V+ S Recolor

28. Case 2 diagram P S V+ P+ S V Recolor and absorb Either extra black absorbed by P or P now has extra blackness

30. Case 3 diagrams P S V+ P S V Rotate P S V Recolor

32. Case 4 Diagrams P S V+ P S V+ Rotate P S V+ Recolor

33. 65 50 80 10 60 70 90 62 Perform the following deletions, in the order specified Delete 90, Delete 80, Delete 70

34. Red Black Trees Top-Down Insertion

39. Possible insertion configurations X (Red or Black) Y Z If a new node is inserted as a child of Y or Z, there is no problem since the new node’s parent is black

40. Possible insertion configurations X Y Z If new node is child of Z, no problem since Z is black. If new node is child of Y, no problem since the new node’s uncle (Z) is black – do a few rotations and recolor…. done

41. Possible insertion configurations X Y Z If new node is inserted as child of Y or Z, it’s uncle will be red and we will have to go back up the tree. This is the only case we need to avoid.

42. Top-Down Traversal X Y Z As we traverse down the tree and encounter this case, we recolor and possible do some rotations. There are 3 cases. Remember the goal – to create an insertion point at which the parent of the new node is Black, or the uncle of the new node is black.

43. Case 1 – X’s Parent is Black X Z Y P X Z P Just recolor and continue down the tree Y

45. Case 2 diagrams X Z Y P G U S X Z Y P G U S Rotate P around G. Recolor X, Y, Z, P and G

47. Case 3 Diagrams (1 of 2) X Z Y P G U S X Y S P G U Z Step 1 – recolor X, Y and Z. Rotate X around P.

48. Case 3 Diagrams (2 of 2) X Y S P G U Z P Y S X G U Z Step 2 – Rotate X around G. Recolor X and G

49. An exercise – insert F D T W Z V L J P E K

50. Top-Down Insert Summary P X Y Z Case 1 P is Black Just Recolor P X Y Z Case 2 P is Red X & P both left/right P X Y Z G P X Y Z G Recolor X,Y,Z P X Y Z G Rotate P around G Recolor P,G Case 3 P is Red X and P are opposite children P X Y Z G Recolor X,Y,Z Rotate X around P X P Y Z G Rotate X around G Recolor X, G Recolor X,Y,Z X P Y Z G

51. Red Black Trees Top-Down Deletion

60. Case 2A X has two Black Children P T X 2A1. T has 2 Black Children 2A2. T’s left child is Red 2A3. T’s right child is Red ** if both of T’s children are Red, we can do either 2A2 or 2A3

61. Case 2A1 X and T have 2 Black Children P T X P T X Just recolor X, P and T and move down the tree

62. Case 2A2 P T X L X has 2 Black Children and T’s Left Child is Red Rotate L around T, then L around P Recolor X and P then continue down the tree L1 L2 P T X L L1 L2

63. Case 2A3 P T X X has 2 Black Children and T’s Right Child is Red Rotate T around P Recolor X, P, T and R then continue down the tree R1 R2 P R X T R2 R1 R L L

65. Case 2B Diagram P X T Move down the tree. P X T P T X If move to Black child (2B2) Rotate T around P; Recolor P and T Back to step 2, the main case If move to the Red child (2B1) Move down again

66. Step 3 Eventually, find the node to be deleted – a leaf or a node with one non-null child that is a leaf. Delete the appropriate node as a Red leaf Step 4 Color the Root Black

67. Example 1 Delete 10 from this RB Tree 15 17 16 20 23 18 13 10 7 12 6 3 Step 1 – Root has 2 Black children. Color Root Red Descend the tree, moving X to 6

68. Example 1 (cont’d) 15 17 16 20 23 18 13 10 7 12 6 3 One of X’s children is Red (case 2B). Descend down the tree, arriving at 12. Since the new X (12) is also Red (2B1), continue down the tree, arriving at 10. X

69. Example 1 (cont’d) 15 17 16 20 23 18 13 10 7 12 6 3 Step 3 -Since 10 is the node to be deleted, replace it’s value with the value of it’s only child (7) and delete 7’s red node X

70. Example 1 (cont’d) 15 17 16 20 23 18 13 7 12 6 3 The final tree after 7 has replaced 10 and 7’s red node deleted and (step 4) the root has been colored Black.

71. Example 2 Delete 10 from this RB Tree 15 17 16 20 13 10 12 6 3 4 2 Step 1 – the root does not have 2 Black children. Color the root red, Set X = root and proceed to step 2

72. Example 2 (cont’d) 15 17 16 20 13 10 12 6 3 4 2 X X has at least one Red child (case 2B). Proceed down the tree, arriving at 6. Since 6 is also Red (case 2B1), continue down the tree, arriving at 12.

73. Example 2 (cont’d) 15 17 16 20 13 10 12 6 3 4 2 X X has 2 Black children. X’s sibling (3) also has 2 black children. Case 2A1– recolor X, P, and T and continue down the tree, arriving at 10. P T

74. Example 2 (cont’d) 15 17 16 20 13 10 12 6 3 4 2 P X T X is now the leaf to be deleted, but it’s Black, so back to step 2. X has 2 Black children and T has 2 Black children – case 2A1 Recolor X, P and T. Step 3 -- Now delete 10 as a red leaf. Step 4 -- Recolor the root black

75. Example 2 Solution 15 17 16 20 13 12 6 3 4 2

76. Example 3 Delete 11 from this RB Tree 15 13 11 12 10 5 7 3 6 9 2 4 Valid and unaffected Right subtree Step 1 – root has 2 Black children. Color Root red. Set X to appropriate child of root (10)

77. Example 3 (cont’d) 15 13 11 12 10 5 7 3 6 9 2 4 X X has one Red child (case 2B) Traverse down the tree, arriving at 12.

78. Example 3 (cont’d) 15 13 11 12 10 5 7 3 6 9 4 X Since we arrived at a black node (case 2B2) assuring T is red and P is black), rotate T around P, recolor T and P Back to step 2 P T 2

79. Example 3 (cont’d) 15 13 11 12 10 5 7 3 6 9 4 X P T 2 Now X is Black with Red parent and Black sibling. X and T both have 2 Black children (case 2A1) Just recolor X, P and T and continue traversal

80. Example 3 (cont’d) 15 13 11 12 10 5 7 3 6 9 4 X P T 2 Having traversed down the tree, we arrive at 11, the leaf to be deleted, but it’s Black, so back to step 2. X and T both have two Black children. Recolor X, P and T. Step 3 -- delete 11 as a red leaf. Step 4 -- Recolor the root black

81. Example 3 Solution 13 12 10 5 7 3 6 9 4 2 15