More Trees

1. MoreTrees

2. Outline BTree, B* Tree, B+ Tree 2-3 Tree, 2-3-4 Tree Red-Black Tree (RBT) Left-Leaning Red-Black Tree Double Red & Double Black

3. B Tree

4. B-Treeisin memory of R. BayerThe B tree is a generalization of a binary search tree in that a node can have more than two children [wiki]. Degree 為 d的 B tree:1) 每個node 包含至多 d 個 child pointers (或 d-1個 elements)2) 每個 node 至少1/2 滿 (即至少[ (d-1)/2]個 elements)

5. 20 28 10 25 30 B-Tree of Degree 3

6. B* Tree B-tree 的 node 至少2/3 滿

7. 20 28 6 15 26 30 10 23 2 4 35 B* Treeof Degree 4

8. B+ Tree • 包含 index pages和 data pages • root node 和 internal nodes 為 index pages (keys only). • leaf nodes 為 data pages (排序的data ) data 即 element (含有 key) • 每個 node 至少 1/2 滿 (Fill Factor 50%).

9. index page data page B+ Tree This B+ tree:

10. 2-3 Tree

11. 2-3 Tree 為search tree 可為空或符合: • 每個 internal node 為 2-node 或 3-node 2-node有2 child pointers, 3-node有3 child pointers • 所有external nodes 都在相同 level. A 2-node 40 B C 3-node internal node 10 20 80 external nodes

12. 40 10 20 70 80 2-3 Tree Insertion insertCase 1:插入 70 • 先尋找 70. 發現不在其中. • 須知尋找70時 遇到哪node. 是 含 80 的 node C • node C 只有一個 element, 所以70 可放 C A B C

13. A 20 40 B D C 10 30 70 80 Figure 3 2-3 Tree insertCase 2: 插入 30 • 會遇到 30 的是 node B • B 為 3-node, 須產生新 node D. • B 含 elements 10, 20, 30 • B中最大element 30 放D • 最小element 10 放 B. • 中間element 20放B的parent A

14. 2-3 Tree Insertion (Cont.) Insert case 3: 插入 60 • 尋找60會遇 node C • C 為 3-node,需產生新 node E • C含 elements 60,70,80 • 中間值 70 放在C的parent A • 最小值 60放C 最大值80放E • A 為 3-node,產生新 node F • A含 elements 20, 40, 70 • 中間值 40 放在A的parent G (需產生 G) • 最小值 20放A 最大值70放F

15. G 40 A F 20 70 B D C E 10 30 80 60 2-3 Tree Insertion (Cont.) Figure4 Insertion of 60 into the 2-3 tree of Figure 3

16. 2-3 Tree Deletion A 50 80 D C B 60 70 10 20 90 95 (a) Initial 2-3 tree A 50 80 D C B 60 10 20 90 95 (b) 70 deleted

17. 2-3 Tree Deletion (Cont.) A 50 80 D C B 60 10 20 95 (c) 90 deleted A 20 80 D C B 50 10 95 (d) 60 deleted

18. 2-3 Tree Deletion (Cont.) A 20 C B 50 80 10 (e) 95 deleted A 20 A C B 20 80 80 10 (g) 10 deleted (f) 50 deleted

19. 2-3-4 Tree

20. 2-3-4 Tree 為 search tree 為空或是滿足: • 每個 internal node 為 2, 3,或 4 node. 2-node 有2 child pointers, 3-node有3 child pointers, 4-node 有4 child pointers • 所有external nodes 都在相同 level. 2-3-4 tree 類似2-3 tree, 但它有 4-node (即 四個 pointers 和三個 elements) 如下圖 50 60 70

21. 2-3-4 Tree Insertion There are 3 cases for a 4-node: Case 1: It is the root Case 2: Its parent is a 2-node Case 3: Its parent is a 3-node (fig. omitted)

22. 2-3-4 Tree Insertion • Case 1: It is the root. t t (root) y x y z x z a b c d a b c d Figure1 when the root is a 4-node

23. 2-3-4 Tree Insertion • Case 2: Its parent is a 2-node z x z e e w x y w y a b c d a b c d Figure 2 when the child of a 2-node is a 4-node

24. 2-3-4 Tree Deletion • Deleting p • The following cases are to be considered: • p is a leaf. • q is not a 2-node. (q is a child of p) • q is a 2-node and its nearest sibling r is a 2-node. • q is a 2-node and its nearest sibling r is a 3-node. • q is a 2-node and its nearest sibling r is a 4-node.

25. 2-3-4 Tree Deletion (Cont.) q is the left child of a 3-node p p p w z x z q f q r r f v x y v w y d e c c d e a b a b Figure1 when the nearest sibling r is 3-node

26. 2-3-4 Tree Deletion (Cont.) q is the left child of a 4-node p p v y z w y z g g q q f f r r u u v w x x d e e c c d a b a b Figure1 when the nearest sibling r is 3-node

27. 2-3-4 Tree 2-3-4 tree 可轉成binary search tree 稱為 red-black tree 可節省儲存空間 因為2-3-4node 會浪費不少未存資料的空的空間

28. Red-Black Tree(RBT)

29. Red-Black Tree red-black tree 為 binarysearch tree: • 每個 node 不是red就是black • 每個leaf (NULL) 都為black • rednode 的兩個children都為black. • 每個 path 含相同數目的 black nodes. • red node不可接著red node (不可紅紅) A basic red-black tree

30. Red-Black Tree A red-black tree with n internal nodes has height at most 2log(n+1). Red-Black tree can always be searched in O(log n) time.

31. S L S L OR c  a L S a c b a b b Red-Black Tree Right- leaning Left-leaning c S for Small; L for Large. Figure 1 Transforming a 3-node into two red_black nodes

32. Red-Black Tree M S M L  S L c d a b a d b c S for Small;M, Middle; L, Large. Figure 2 Transforming a 4-node into two red_black nodes

33. 50 10 70 7 40 60 80 75 90 5 9 30 85 92 將下圖的 Red-Black Tree 轉成 2-3-4 Tree 依序 (1)刪除60 (2)加入8再轉回 Red-Black Tree

34. 50 10 70 5 30 60 75 85 80 7 40 90 9 92 上圖轉成的2-3-4 Tree

35. 刪除 60 70

36. 加入 8 8 7

37. 轉回Red-Black Tree

38. Left-Leaning Red-Black Tree

39. LLRBT is easier to implement than RBT, especially the deletion.It requires 3-nodes are left-leaning, thus maintains 1-1 correspondence with 2-3-4 trees(see next page).

40. L S L c  S a c b a b LL Red-Black Tree Left-leaning S for Small; L for Large. Transforming a 3-node into LL red black nodes

41. Double Red & Double Black

42. During Red-Black Tree Insertion, Double Red may occur as shown next.

43. Red-Black Tree Insertion 我們要對左圖　insert 4 1)依 binary search tree 　　把 4 當 3 的 right child 2)依 red black tree 新加入者為red 故 3,4 形成右圖 Double Red違反 Red Rule 2 2 1 3 3 1 4

44. Double Red in 2-3-4 Tree 1 1 2 2 3 3 已滿, 此時 insert 4 這 node 爆掉了,故要調整之 4 對應的 Red-Black Tree: 2 3 4 此時 叫 Double Red雙紅, 表示原來 node 爆掉了 1 3 4

45. During Red-Black Tree Deletion, Double Black may occur as shown next.

46. Double Black in 2-3-4 Tree 7 5 3 7 3 8 8 此時Delete 5 對應的 Red-Black Tree: 7 8 Double BLACK 雙黑線,表示 其中有個空 2-3-4 node.故要調整之 3