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Red Black Tree Rules and Properties Explanation

Learn about the rules and properties of Red-Black trees, including node color, rotation, and uncle cases. Understand how to maintain balance and adherence to the Red-Black tree principles.

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Red Black Tree Rules and Properties Explanation

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  1. Red Black Tree 5 Rules

  2. Red Black Tree Properties - A • Every NILL Node Is BLACK  sometimes referred to as a "Null Leaf" • Every Node Is Either RED or BLACK

  3. Red Black Tree Properties - B • Every RED Node Has Two Black Child Nodes (Or Is Childless) There Can't Be Two RED Nodes In A Row • Every Distance Path From Node X Down To A Leaf Node Has The Same Number Of BLACK Nodes • The Root Node Is Always BLACK

  4. Rotation GeneralizationRight Rotation

  5. Right Rotation - 1/2 • There Exists Some Square Node (somewhere in the tree) • There Exists Some Circle Node (Where Value In Circle Is Less Than The Value In Square) C • There Exists Some Subtrees: A, B, & C A B • Putting Values In Nodes Is Not Necessary, But Since I Did It To Demonstrate The AVL Rotations, I Will Do It For The Red Black.

  6. Right Rotation - 2/2 100 90 C 90 100 A • Simple Way To Rotate The Valid Binary Search Tree Without Altering The Search Capabilities. A B C B • Single Rotate Right  AVL

  7. Rotation GeneralizationLeft Rotation

  8. Left Rotation - 1/2 • There Exists Some Square Node (somewhere in the tree) • There Exists Some Circle Node (Where Value In Circle Is Greater Than The Value In Square) A • There Exists Some Subtrees: A, B, & C C B • Putting Values In Nodes Is Not Necessary, But Since I Did It To Demonstrate The AVL Rotations, I Will Do It For The Red Black.

  9. Left Rotation - 2/2 90 100 A 100 C 90 C B A B • Simple Way To Rotate The Valid Binary Search Tree Without Altering The Search Capabilities. • Single Rotate Left  AVL

  10. Violation 1LRED UNCLE Case(On Left)

  11. Pg Pf Pu P Easy Case 1L - RED UNCLE Case - 1 (P On Left) a • This RED UNCLE Case Is Occurring As We Walk Our Way Up The Tree. 125 • Label P Pf  Pg  Pu • If We Make The New Node RED, It Minimizes The Number Of Bad Things That Can Happen: 100 e d f 150 50 • The Color Distance Will Not Change The Number of Black Down Any Branch b c • Only Thing That Can Go Wrong Is Two Red Nodes In A Row • Label The Distance Paths

  12. Easy Case 1L - RED UNCLE Case - 2 (P On Left) • Record The Relative Distances Down The Various Paths In The Tree: a 125 • Easy Case 1L: UNCLE is RED  No Rotation Required! • ab = 1 • ac = 1 100 150 • ad = 1 • ae = 1 e d f 50 • af = 1 b c

  13. Easy Case 1L - RED UNCLE Case - 3 (P On Left) a a Pg 125 • How Fix? Pg 125 • Easy Case 1L: UNCLE is RED  No Rotation Required! Pf 100 150 Pu Pf 100 150 Pu • ab = 1 e d • ac = 1 f 50 e d P f 50 • ad = 1 P • ae = 1 • af = 1 b c b c • Pf  Color = BLACK Pu  Color = BLACK Pg Color = RED • Double Check The Distances ab=1, ac=1, ad=1, ae=1, af=1  YEA!

  14. Case 1L Rule Summary a a Pg 125 Pg 125 Pf 100 150 Pu Pf 100 150 Pu e d f 50 e d P f 50 P b c b c

  15. Violation 1RRED UNCLE Case(On Right)

  16. Pg Pu Pf P Easy Case 1R - RED UNCLE Case - 1 (P On Right) a • This RED UNCLE Case Is Occurring As We Walk Our Way Up The Tree. 150 • Label P Pf  Pg  Pu • If We Make The New Node RED, It Minimizes The Number Of Bad Things That Can Happen: 125 b c 200 d 50 • The Color Distance Will Not Change The Number of Black Down Any Branch e f • Only Thing That Can Go Wrong Is Two Red Nodes In A Row • Label The Distance Paths

  17. Pg Pu Pf P Easy Case 1R - RED UNCLE Case - 2 (P On Right) a • Record The Relative Distances Down The Various Paths In The Tree: 150 • Label P Pf  Pg  Pu • Easy Case 1R: UNCLE is RED  No Rotation Required! • ab = 1 • ac = 1 125 • ad = 1 • ae = 1 b c 200 d 50 • af = 1 e f • Label The Distance Paths

  18. Easy Case 1R - RED UNCLE Case - 3 (P On Right) a a Pg 150 • How Fix? Pg 150 • Easy Case 1L: UNCLE is RED  No Rotation Required! 125 Pu Pf 125 Pu Pf • ab = 1 • ac = 1 b 200 200 P 50 c d b c 50 P d • ad = 1 • ae = 1 • af = 1 e f e f • Pf  Color = BLACK Pu  Color = BLACK Pg Color = RED • Double Check The Distances ab=1, ac=1, ad=1, ae=1, af=1  YEA!

  19. Case 1R Rule Summary a a Pg 150 Pg 150 125 200 Pu Pf 125 200 Pu Pf b P 50 c d b c 50 P d e f e f

  20. RED Uncle Can Occur With InsertionRED UNCLE Case(On Left)

  21. Easy Case 1L - RED UNCLE Case - 1 (P On Left) • Distance Property = 2 For All - If Count Null • Add 50 to Tree - Make it Red • Minimize The Number Of Bad Things That Can Happen: 200 • Easy Case 1L: UNCLE is RED  No Rotation Required! • Will Not Change The Number of Black Down Any Branch 250 Pg 125 • Only Thing That Can Go Wrong Is Two Red Nodes In A Row 50 150 100 Pf Pu P • Label P Pf  Pg  Pu

  22. Easy Case 1L - RED UNCLE Case - 2 (P On Left) • How Fix? • Might Become Easier If We Don't Try To Fix Everything At OnceStart With This Portion 200 • Easy Case 1L: UNCLE is RED  No Rotation Required! Pg 250 125 • THIS IS EASY CASE - Pu Is Red (125) 50 150 100 Pf Pu • Make Pg REDMake Pf BLACKMake Pu BLACK P • This Shifts The Problem Up Two Generations!

  23. Easy Case 1L - RED UNCLE Case - 3 (P On Left) • How Fix? • How Fix? • Change Pf To BLACK Pf 200 250 125 P • Distance Property = 3 For All 50 150 100 • 2 BLACK UNCLE Cases: One Requires A RotationOne Requies A Preliminary Rotation And A Second Rotation

  24. RED Uncle Can Occur With InsertionAdd To Tree Violation 1RRED UNCLE Case(On Right)

  25. Easy Case 1R - RED UNCLE Case - 1 (P On Right) • Distance Property = 2 For All If Count Null • Add 225 to Tree - Make it Red 100 • Minimize The Number Of Bad Things That Can Happen: • Easy Case 1R: UNCLE is RED  No Rotation Required! 50 150 Pg • Will Not Change The Number of Black Down Any Branch • Only Thing That Can Go Wrong Is Two Red Nodes In A Row 125 Pu Pf 225 200 P • Label P Pf  Pg  Pu

  26. Easy Case 1R - RED UNCLE Case - 2 (P On Right) • How Fix? 100 • Might Become Easier If We Don't Try To Fix Everything At OnceStart With This Portion 50 150 Pg • THIS IS EASY CASE - Uncle Of P Is Red (125) 125 Pu Pf 200 225 • Make Pg REDMake Pf BLACKMake Pu BLACK P • This Shifts The Problem Up Two Generations!

  27. Easy Case 1R - RED UNCLE Case - 3 (P On Right) • How Fix? Pf 100 • Change Pf To BLACK 50 P 150 • Distance Property = 3 For All 125 200 225 • 2 BLACK UNCLE Cases: One Requires A RotationOne Requies A Preliminary Rotation And A Second Rotation

  28. Violation 3LBLACK UNCLE CaseWith One Rotation (On Left)

  29. Case 3L - BLACK UNCLE One Rotation Class- 1 (P On Left) a • This BLACK UNCLE Case Is Occurring As We Walk Our Way Up The Tree. 200 • We Have Already Established That The Distance Property Is OK 250 125 • ab = 1 e f d • ac = 1 100 • ad = 1 • ae = 2 b c • af = 2 • GOAL: Fix The RED CHILD With RED Father Problem Without Messing Up The Distance Properties.

  30. Case 3L - BLACK UNCLE One Rotation Class- 2 (P On Left) a a Pg 200 125 250 Pu 125 Pf 100 Pf • ab = 1 e f d • ac = 1 Pg 100 d 200 P • ad = 1 • ae = 2 P • af = 2 b c 250 Pu b c • Solve RED RED Problem? • Promote Pf Up • Hook up 'd' e f • Label P Pf  Pg  Pu • Hook Pg Right of Pf • Swap Colors Pg Pf • Double Check The Distances ab=1, ac=1, ad=1, ae=2, af=2  YEA!

  31. Case 3L Rule Summary a a Pg 200 125 Pf 250 Pu 125 Pf P Pg 100 200 e f d Pu b c 100 d 250 P b e c f

  32. Violation 3RBLACK UNCLE CaseWith One Rotation (On Right)

  33. Case 3R - BLACK UNCLE One Rotation Class- 1 (P On Right) a • This BLACK UNCLE Case Is Occurring As We Walk Our Way Up The Tree. 200 • We Have Already Established That The Distance Property Is OK 250 125 • ab = 2 d c b • ac = 2 300 • ad = 1 f e • ae = 1 • af = 1 • GOAL: Fix The RED CHILD With RED Father Problem Without Messing Up The Distance Properties.

  34. Case 3R - BLACK UNCLE One Rotation Class- 2 (P On Right) a a Pg 200 Pf 250 Pu 250 Pg 125 Pf P 300 200 • ab = 2 d e b c f 300 Pu P d • ac = 2 125 • ad = 1 • ae = 1 e f • af = 1 b c • Solve RED RED Problem? • Promote Pf Up • Hook up 'd' • Label P Pf  Pg  Pu • Swap Colors Pg Pf • Hook Pg Left of Pf • Double Check The Distances ab=2, ac=2, ad=1, ae=1, af=1  YEA!

  35. Case 3R Rule Summary a a Pg 200 Pf 250 Pg Pu 250 125 Pf P 300 200 d Pu d e b c f 300 P 125 e f b c

  36. Violation 2LBLACK UNCLE CaseWith Two Rotations (On Left)

  37. Case 2L - BLACK UNCLE Two Rotations Class- 1 (P On Left) a • This BLACK UNCLE Case Is Occurring As We Walk Our Way Up The Tree. Pg 200 • We Have Already Established That The Distance Property Is OK 250 Pu 125 Pf • ab = 1 b • ac = 1 e f P 150 • ad = 1 • ae = 2 c d • af = 2 • Label P Pf  Pg  Pu • Case 2L is going to turn into Case 3L. • This Is The Worst Possible Case

  38. Case 2L - BLACK UNCLE Two Rotations Class- 2 (P On Left) 1 a a Pg 200 Pg 200 250 Pu 100 Pf 250 Pu 100 Pf • ab = 1 b e f P b 125 e • ac = 1 f c P 125 • ad = 1 • ae = 2 c • af = 2 d d • Hook P To The Left Of Pg • Might Become Easier If We Don't Try To Fix Everything At OnceStart With This Portion • Hook Pf To The Left Of P • Hook up 'c' • This Is Case 3L

  39. Case 2L - BLACK UNCLE Two Rotations Class- 3 (P On Left) 1 a Pg 200 250 Pf Pu 125 P 100 • ab = 1 b e • ac = 1 f c d • ad = 1 • ae = 2 • af = 2 • Swap Pointers Pf  P

  40. Case 2L - BLACK UNCLE Two Rotations Class- 4 (P On Left) 2 a a 125 Pg 200 250 Pf Pu 125 P 100 100 Pf • ab = 1 b e • ac = 1 f c d Pg d 200 • ad = 1 • ae = 2 P • af = 2 250 Pu • Swap Pointers Pf & P b c e f • Do Case 3L Rotation • Double Check The Distances ab=1, ac=1, ad=1, ae=2, af=2  YEA!

  41. Case 2L Rule Summary a a Pg 200 150 P 250 Pu 125 Pf Pg 125 Pf 200 b e f P 150 b c d 250 Pu c d e f

  42. Violation 2RBLACK UNCLE CaseWith Two Rotations (On Right)

  43. Case 2R- BLACK UNCLE Two Rotations Class- 1 (P On Right) a a • This BLACK UNCLE Case Is Occurring As We Walk Our Way Up The Tree. Pg 200 • We Have Already Established That The Distance Property Is OK Pu 300 125 Pf • ab = 2 • ac = 2 b c P 250 f • ad = 1 • ae = 1 d e • af = 1 • Case 2R is going to turn into Case 3R. • This Is The Worst Possible Case

  44. Case 2R - BLACK UNCLE Two Rotations Class- 2 (P On Right) 1 a a a Pg 200 Pg 200 Pu 300 125 125 Pu Pf P 250 • ab = 2 f P b c 250 b c • ac = 2 d 300 Pf • ad = 1 • ae = 1 d e • af = 1 e f • Might Become Easier If We Don't Try To Fix Everything At OnceStart With This Portion • Hook P To The Right Of Pg • Hook Pf To The Right Of P • Hook up 'e' • This Is Case 3L

  45. Case 2R - BLACK UNCLE Two Rotations Class- 3 (P On Right) a Pg 200 250 125 Pu Pf • ab = 2 b c d • ac = 2 P 300 • ad = 1 • ae = 1 • af = 1 e f • Swap Labels Pf  P • This Is Case 3L

  46. Case 2R - BLACK UNCLE Two Rotations Class- 4 (P On Right) 2 a a Pf 250 Pg 200 P 300 200 250 125 Pu Pf • ab = 2 e f 125 d b c d • ac = 2 P 300 • ad = 1 • ae = 1 b c • af = 1 e f • Swap Labels Pf  P • Do Case 3R Rotation • Double Check The Distances ab=2, ac=2, ad=1, ae=1, af=1  YEA!

  47. Case 2R Rule Summary a a a P 250 Pg 200 Pg Pf 300 200 Pu 300 Pf 125 e f 125 Pu d b c P 250 f b c d e

  48. Rotation GeneralizationRight Rotation

  49. Practice Problem#1

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