Old Trees 1 Outline Tree BTree 2 3

Old Trees 1

Outline • • • Tree B-Tree 2 -3 -4 Tree Red-Black Tree 2

Tree – 從 root 開始存取. – node 為 leaf (external node)或 internal node. • internal node (parent)有一或多個 child nodes, • 相同 parent 的 nodes 互稱為siblings. 3

Tree Root Node Parent Node Child Node leaf Internal node Node leaf 4

B-Tree in memory of R. Bayer 5

B-Tree – B tree 中 degree 為 d 的node 含 至多 d 個 pointers 及 d-1 elements – 每個 node 至少 1/2 滿 至少含floor ( (d-1)/2 ) elements. 6

B-Tree This is a B-Tree of degree 3 20 10 28 25 30 7

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B* Tree B-tree 的node 至少 2/3 full 重分配 elements, 填滿 2 child nodes, 然後分成 3個 nodes 9

B* Tree This is a B*-Tree of degree 4 6 2 4 10 20 15 28 23 26 30 35 10

B+ Tree Data 排序儲存在 leaves. • B+Tree 包含 index pages 和 data pages • root 和中間 nodes為index pages (keys only). • leaf nodes為data pages. • 每個 node 至少 1/2 full. 11

B+ Tree index page data page This examples conforms to: Number of Keys 4 Number of Pointers 5 Fill Factor 50% Minimum Keys in each page 2 12

B+-Tree insert an element 時, 須考慮 3 cases: Leaf Page Full Index Page Full No No Yes Yes 13

B+-Tree • Case 1: 插入一 element到非滿的leaf page 把 element 放sorted leaf page 的適當位置 14

B+ Tree Case 2: 加入 element 時 leaf page 為滿, 但 index page 沒滿 1. 分裂leaf page 2. 把middle 放已排好的index page 3. left leaf page : 比middle 小的 elements 4. right leaf page : 比middle 大的 elements. 15

B+-Tree • 插入 70 到B+ tree. • 70 將加入到 leaf page: (50, 55, 60, 65) • 不幸此 page已滿. 我們須分裂( split ) page 如下: Left Leaf Page Right Leaf Page 50 55 60 65 70 16

B+-Tree • middle 60 放在 50和75之間的 index page • 下面表示加入 70後的B+ tree 17

B+-Tree Case 3: 加入element 當 leaf page 和 index page 都滿了 1. 2. 3. 4. 5. 6. 7. 分裂 leaf page. < middle 放在 left leaf page. >= middle 放在right leaf page. 分裂 index page. elements < middle 放在 left index page. elements > middle 放在 right index page. middle 放在上一層 index. 如上一層 index也滿了, 持續分裂 index pages (steps 4 - 7). 18

B+-Tree • 加入 95 到 B+ tree. • 95將加入 (75, 80, 85, 90). 既然此page 已滿 , 分裂: Left Leaf Page Right Leaf Page 75 80 85 90 95 • middle 85, 升到 index page. 不幸地, index page 也滿 了, 所以分裂 index page: Left Index Page Right Index Page New Index Page 25 50 75 85 60 19

B+-Tree • Deletion 須考慮 3 Cases: Leaf Page Below Fill Factor Index Page Below Fill Factor Case 1 No No Case 2 Yes No Yes Case 3 21

B+-Tree Case 1: delete 後, leaf page 和 index page 不符合 B+tree – 從 leaf page delete此element – 把element安排成升冪排序. – 如刪除的element在 index page, 用下一element來取代它 22

B+-Tree • Delete 70 – 70 is in a leaf page containing 60, 65 and 70. This page will contain 2 elements after the deletion. Since our fill factor is 50% (2 elements), simply delete 70 from the leaf node. 70 23

B+-Tree Case 2: delete後, index page 不夠滿 – – 結合 leaf page 和它的 sibling. 改變 index page 來反應 leaf page 的改變 24

B+-Tree • Delete 25 – 25 is found in the leaf node containing 25, 28, and 30. The fill factor will be 50% after the deletion; however, 25 appears in the index page. Thus, when we delete 25, we must replace it with 28 in the index page. 25

B+-Tree Case 3: Delete後 , index page 和 data page 不夠滿 1. 結合(merge) leaf page 和它的 sibling. 2. 調整 index page 來反應其改變 3. 結合 index page 和它的 sibling. 持續結合 index pages 直到符合tree要求, 或到達 root page. 26

B+-Tree • Delete 60 1. 2. 3. The leaf page containing 60 (60 65) will be below the fill factor after the deletion. Thus, we must merge leaf pages. With merged pages, the index page (75, 85) will be reduced by one key. Hence, it (85)will fall below the fill factor. Thus, we must merge index pages. 60 is now the only key in root index page. Obviously, it will be removed. 27

2 -3 Tree 28

2 -3 Tree • 2 -3 tree為 search tree 可為空 或: 每個 internal node 為 2 -node 或 3 -node: 2 -node 有兩個 children 3 -node 有三個 children – Fig. 1: element 40 A B 10 20 C 80 29

2 -3 Tree 23 Node data structure: public class Tree 23 Node { private element. Type left. Item; private element. Type right. Item; private Tree 23 Node left. Child; private Tree 23 Node middle. Child; private Tree 234 Node right. Child; // constructors omitted // } 30

• 插入 2 -3 tree 2 -3 Tree – Case 2: 插入 30 到 Figure 2. • 會遭遇到 30的leaf node 為node B • 既然 B 為 3 -node, 須產生新 node D. • D 將含B 中最大element B • 最小的element 會在 B. • 中間的element會在B的parent 10 A A 20 40 D 30 Figure 3 C 70 80 32

2 -3 Tree A 50 80 B 10 20 D C 60 70 90 95 (a) Initial 2 -3 tree A 50 80 B 10 20 D C 60 90 95 (b) 70 deleted 33

2 -3 Tree A 50 80 B 10 20 D C 60 95 (c) 90 deleted A 20 80 B 10 D C 50 95 (d) 60 deleted 34

2 -3 Tree A 20 B 10 C 50 80 (e) 95 deleted A 20 B 10 C A 20 80 80 (f) 50 deleted (g) 10 deleted 35

2 -3 -4 Tree 36

2 -3 -4 Tree • 2 -3 -4 tree 為search tree 為空或: 每個 internal node 為 2, 3, 或 4 node. 2 -3 -4 tree 類似一個 2 -3 tree, 但它允許: 4 -node: 四個 children 和三個 elements 50 60 70 37

2 -3 -4 Tree 2 -3 -4 tree node data structure: public class Tree 234 Node { private element. Type small. Item; private element. Type middle. Item; private element. Type large. Item; private Tree 234 Node leftchild; private Tree 234 Node lmidchild; private Tree 234 Node rightchild; // constructors omitted // } 38

2 -3 -4 Tree Insertion • Insertion to a 2 -3 -4 tree – There are essentially 3 cases to consider 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 39

2 -3 -4 Tree Insertion • Case 1: t t (root) y x y z x a b c z d a b c d Figure 1 when the root is a 4 -node 40

2 -3 -4 Tree Insertion • Case 2: z w a b x y c x e d z w a b e y c d Figure 2 when the child of a 2 -node is a 4 -node 41

2 -3 -4 Tree Deletion • We are now at node p and will move to node q NEXT. Consider the following 4 CASES: (1) p is a leaf: delete the element in p OR it is NOT in the tree. (2) q is not a 2 -node : In this case, the search moves to q (3)q is a 2 -node and its nearest sibling r is also a 2 -node. (3. 1) if p is a 2 -node, merge p, q, r to a 4 -node. (3. 2) if p is a 3 or 4 -node, merge q, r (4) q is a 2 -node and its nearest sibling r is NOT a 2 -node : see fig. in case 4. 42

2 -3 -4 Tree Deletion • Case 1: v 在 leaf 直接刪除之 a p v w b a 刪除v f r x y p w b r f x y 43

2 -3 -4 Tree Deletion • Case 2: q是 3 -node，p移到q時找到要刪除的v，則刪除之 p w q s z w r v p f x f r s y z x y 刪除v a t g c d e 44

2 -3 -4 Tree Deletion • Case 3. 1: merge 三個 2 -node p p w q v a b v w x r x c a b c d d 45

2 -3 -4 Tree Deletion Case 3. 2: w 下移到q merge q, r 便可刪除v p w q z z f r v a p q x b c v d a w b x d f e 46

2 -3 -4 Tree Deletion Case 4: w下移到q，x上移到p 若找到欲刪除的v，便可刪之 p q z x f r v a w p x b c q y d v e a z r w b c f y d e 47

2 -3 -4 Tree to Red-Black Tree 2 -3 -4 tree 可轉換成特殊的 binary search tree 稱為 red-black tree 可節省儲存空間 48

Red-Black Tree 49

Red-Black Tree red-black tree 兩特性: Path rule: • 各path 從root 到leaf (或有一 子的node)有同數目的 black nodes. Red Rule: • red node的孩子不可為red node A basic red-black tree 50

Red-Black Tree S L OR c a L S a b c a b b c S for Small; L for Large. Figure 1 Transforming a 3 -node into two red_black nodes 51

Red-Black Tree M SML a b c S d a L b c d S for Small; M, Middle; L, Large. Figure 2 Transforming a 4 -node into two red_black nodes 52

範例： 將下圖的 Red-Black Tree 轉成 2 -3 -4 Tree 再依序 (1)刪除 60 (2)加入 8 畫出 2 -3 -4 Tree ，再轉成 Red-Black Tree 。 50 10 7 5 70 40 9 30 60 80 75 90 85 92 53

練習： 將下圖的 Red-Black Tree 轉成 2 -3 -4 Tree 再依序 (1)刪除 60 (2)加入 8 畫出 2 -3 -4 Tree ，再轉成 Red-Black Tree 。 50 10 7 5 70 40 9 30 60 80 75 90 85 Figure 3 Red-black 92 54

2 -3 -4 Tree 50 10 5 7 9 30 40 70 80 60 75 85 90 92 Figure 4 2 -3 -4 tree 55

刪除 60 70 56

加入 8 7 8 57

轉成 Red-Black Tree 50 7 80 5 10 70 30 8 9 90 75 85 92 40 58