Binary Search Trees

1
Binary Search Trees

• ??????? ???? ?? ? ???????? ??

2
???? ?????? Binary Trees

• ????? ???????
• ???? ???? ????? ?????? ???
• ???? ?? ??????? ??? ?? ?????? ?????? ?????
  ?????? ???
• ????? ?????? ??? ?? ?? ???? ?? 1 ? ?? ????? 2? ??
  ????? ?????? ????? ?? ?? ????.

56
??? ???? ?? ???? ??? ?
3
?????? ???? ??????

• ?????? ???? ????? ?? ??? ( ??? ??? ?????) ?? ???
  ?????? ????
• ??????? Recursive
• ?????? Iterative
• ?????? ??????? ?????? ??? ? ?????? ??? (???????)
  ?? ? ???? ??
• Inorder 1-????? ??? ??? 2-???? ? 3- ????? ???
  ????
• Preorder 1- ????? 2- ????? ??? ??? 3- ????? ???
  ????
• Postorder1-????? ??? ??? 2- ????? ??? ????? 3-
  ????

4
?????? ???????? ????

• (a b) (c d) e f/gh 3.25
• ?? ????? ????? ?? ?? ??? ??? ??? ???
• ??????.Operators (, -, /, ).
• ?????? ??
• Operands (a, b, c, d, e, f, g, h, 3.25, (a b),
  (c d), etc.).
• ????? ???? ???? ???? ? Delimiters ((, )).

5
???? ?? ?????

• ????? ?????????? ?? ?? ????? ???? ????
• ????? ??? ?? ???? ? ??????? ??????? ???? ???
• ???????? ??????? ?? ?????? ???? ?????
• a b
• c / d
• e - f
• ???????? ?????? ?? ?????? ???? ?????
• g
• - h
• ????? ???????? ????
• ????? ????????
• ????? ???????
• ????? ??????

6
????? ????????

• ?????? ??? ?? ????? ???? ?? ????
• a b
• a b c
• a b / c
• (a b) (c d) e f/gh 3.25
• ????? ? ????? ???????
• ???? ????? ???? ????? ???? ???
• ?????? ??? ? ????? ?? ??? ? ????? ???? ?????
• ?? ?? ?? ???? ???

7
????? ???? ????

• ?? ??????? ?? ??? ?????? ????????? ????? ?? ????
  ?? ?? ?????? ?????? ??? ?? ?????? ???? ?? ????
• (a b) (c d) / (e f)

8
?????? ?????

• ???? ?????? ???????? ???? ?? ????? ???? ?????
  ????? ???? ??????? ???? ???
• ???? ?????? ???? ?????? ?????? ?? ???? ?????
• ????? ??????? ? ?????? ???? ???????? ??????
• ?????? ?????? ?? ??? ?????? ?? ??????? ????
  ???????? ?????? ???
• ????? ?????? Postfix
• ????? ?????? ?? ????? ?????? ?? ?? ?????? ?? ????
  ??? ???
• A o B AB o ,
• o is the operator
• ????? ??????? ?? ????? ????? ???? ???
• A o B oAB ,
• o is the operator

9
??? ????? ????? ??????

• Infix a b c
• Postfix

a
b
c

• Infix a b c
• Postfix

a
b

c

• Infix (a b) (c d) / (e f)
• Postfix

a
b

c
d
-

e
f

/
10
???????? ?????

• ???? ??????? ????? ?? ?????? ???? ???? ????????
  ????? ??????? ?? ????
• a gt a _at_
• a b gt a _at_ b
• - a gt a ?
• - a-b gt a ? b -

11
Postfix Evaluation

• ????? ?? ?? ?? ?? ???? ?? ?????? ? ?? ?? ??????
  ?? ?? ???? ?? ?? ?? ???? ??????? ?? ????
• ?? ????? ?? ????? ?? ????? ???? ?????? ?? ???? ??
  ?? ????? ? ??? ?????? ??? ?? ????? ?? ???? ? ???
  ????? ?????? ?? ?? ???? ??????? ?? ????
• ????? ????? ????? ?? ????? ???? ???? ????

12
Postfix Evaluation

• (a b) (c d) / (e f)
• a b c d - e f /
• a b c d - e f /

• b c d - e f /

• c d - e f /

b
a
13
Postfix Evaluation

• (a b) (c d) / (e f)
• a b c d - e f /
• a b c d - e f /

d
c

• c d - e f /

(a b)

• d - e f /

• - e f /

14
Postfix Evaluation

• (a b) (c d) / (e f)
• a b c d - e f /

• e f /

(c d)
(a b)
15
Postfix Evaluation

• (a b) (c d) / (e f)
• a b c d - e f /

• a b c d - e f /

• a b c d - e f /

• a b c d - e f /

f
e

• a b c d - e f /

(a b)(c d)
stack
16
Postfix Evaluation

• (a b) (c d) / (e f)
• a b c d - e f /

• a b c d - e f /

• a b c d - e f /

• a b c d - e f /

(e f)

• a b c d - e f /

(a b)(c d)

• a b c d - e f /

stack
17
????? ?? ????? ?? ???? ??????

• a b

• - a

18
????? ?? ????? ?? ???? ??????

• (a b) (c d) / (e f)

19
?????? ????? ?? ????

• ????????? ???? ? ?? ?? ?????? ?? ???? ???? ???
• ?? ???????? ??? ??????? ?????? ?? ???? ?????? ??
  ?????? ???

20
?????? ????

• ?????? ???? ?????? ?????? ??
• ?? ??? ?? ??? ????? ?? ?????
• ????? ???? ?? ??? ??? ?????? ??? ?? ??? ?? ?????
  ?? ????
• Preorder
• Inorder
• Postorder
• Level order

21
?????? ??? ?????

• public static void preOrder(BinaryTreeNode t)
• if (t ! null)
• visit(t)
• preOrder(t.leftChild)
• preOrder(t.rightChild)

22
Preorder Example (visit print)
a
b
c
23
Preorder Example (visit print)
a
b
d
g
h
e
i
c
f
j
24
Preorder Of Expression Tree
/


a
b
-
c
d

e
f
Gives prefix form of expression!
25
Inorder Traversal

• public static void inOrder(BinaryTreeNode t)
• if (t ! null)
• inOrder(t.leftChild)
• visit(t)
• inOrder(t.rightChild)

26
Inorder Example (visit print)
b
a
c
27
Inorder Example (visit print)
g
d
h
b
e
i
a
f
j
c
28
?????? ???? ????? ?? ????? ????? ????
29
Inorder Of Expression Tree
?????? ???? ?????? ????? ??????? ?? ???? ??????
?? ???
30
Postorder Traversal

• public static void postOrder(BinaryTreeNode t)
• if (t ! null)
• postOrder(t.leftChild)
• postOrder(t.rightChild)
• visit(t)

31
Postorder Example (visit print)
b
c
a
32
Postorder Example (visit print)
g
h
d
i
e
b
j
f
c
a
33
Postorder Of Expression Tree
a
b

c
d
-

e
f

/
Gives postfix form of expression!
34
Traversal Applications

• Make a clone.
• Determine height.
• Determine number of nodes.

35
Level Order

• Let t be the tree root.
• while (t ! null)
• visit t and put its children on a FIFO queue
• remove a node from the FIFO queue and call it
  t
• // remove returns null when queue is empty

36
Level-Order Example (visit print)
a
b
c
d
e
f
g
h
i
j
37
???? ???? ??????

• ??? ???? ????? ???? ??? ?????? ?????
• ??? ?? ????? ??? ?? ????????? ?? ???? ???? ??
  ????!?
• ??? ?????? ???? ??? ??? ?? ?? ??? ????? ????? ???
  ???? ???? ?? ?? ???? ????? ???? ????
• ??? ?? ????? ?? ??? ?????? ?? ???? ?? ???? ?? ??
  ???? ????? ???? ???? ?
• ???? ???????!

38
Some Examples

• preorder ab

inorder ab
postorder ab
level order ab
39
Preorder And Postorder

• preorder ab

postorder ba

• Preorder and postorder do not uniquely define a
  binary tree.
• Nor do preorder and level order (same example).
• Nor do postorder and level order (same example).

40
Inorder And Preorder

• inorder g d h b e i a f j c
• preorder a b d g h e i c f j
• Scan the preorder left to right using the inorder
  to separate left and right subtrees.
• a is the root of the tree gdhbei are in the left
  subtree fjc are in the right subtree.

41
Inorder And Preorder

• preorder a b d g h e i c f j
• b is the next root gdh are in the left subtree
  ei are in the right subtree.

42
Inorder And Preorder

• preorder a b d g h e i c f j
• d is the next root g is in the left subtree h
  is in the right subtree.

43
Inorder And Postorder

• Scan postorder from right to left using inorder
  to separate left and right subtrees.
• inorder g d h b e i a f j c
• postorder g h d i e b j f c a
• Tree root is a gdhbei are in left subtree fjc
  are in right subtree.

44
Inorder And Level Order

• Scan level order from left to right using inorder
  to separate left and right subtrees.
• inorder g d h b e i a f j c
• level order a b c d e f g h i j
• Tree root is a gdhbei are in left subtree fjc
  are in right subtree.

45
???? ??????? ?? ????

• ???? ??????? ?? ???? Binary Search Tree (BST)
• ??????? ???? ?? ???? ???? ??????? ?????? ???
  ?????? ?? ?????? ??? ?? ????? ?? ???
• Search, Minimum, Maximum, Predecessor, Successor,
  Insert, and Delete.
• ???? ????? Dictionary. ? Priority Queue ?? ????
  ?? ?? ??????? ???
• ???? ????? ?????? ?????? ?????? ?? ???(??????)
  ???? ???. O(h).

46
?????BST

• ?? ??????? ?? ?????? ?????? ????? ????? ???? ??
  ???
• root(T) T ????? ?? ?? ???? ????
• ?? ??? ????? ??? ??? ?? ????
• key
• left pointer to left child root of left
  subtree.
• right pointer to right child root of right
  subtree.
• p pointer to parent. prootT NIL
  (optional).

47
???? BST

• ????? ???? (key) ???? ??? ???? ?? ????? ????
• ? y in left subtree of x, then keyy ? keyx.
• ? y in right subtree of x, then keyy ? keyx.

56
48
??????Inorder
?? ?????? ?? ????? BST ?? ???? ?????? ???? ?? ??
???? ????? ????? ???? ???

• Inorder-Tree-Walk (x)
• 1. if x ? NIL
• 2. then Inorder-Tree-Walk(leftp)
• 3. print keyx
• 4. Inorder-Tree-Walk(rightp)

56
12
18
24
26
27
28
56
190
210
213

• ????? ?????? ???? ??? ?

49
????? ?????? Inorder

• ????? ????? ? ????? ?????
• ?? ??????? ?? ??????? ????? ?????? ???? ??? ??
  ? ???? ?? ???? ????? ?? ???
• ??? ???? ??? ????? ????? ? ????? ????? ????? ???
• ??? ???? ??? ?? ?? ??? ????
• ?????? ??? ?? - ?? ??????? ?? ??????? - ??? ??
  ???
• ???? ??? ?? ???? ?? ???? ?? ????? BST? ????? ????
  ?? ?????? ?????? ??? ?? ?????? ??? ???????? ??
  ?????? ??? ??? ?? ???
• ???? ??? ???? ??? ????? ??????? ?????? ??? ?? ???
  ? ?????? ?? ?? ????? ???? ????????

50
????? ?? BST

• ???? ?????? ?????? ??? ???? ?? ?? ???? ?? ?????
  ?? ?????? ?? ??? ???? ????? ??? O(h).
• ??? ???? ?????? ????( ??? ?????? ??? ??? ??? ????
  ????? ?? ????) h ?(lg n)
• ?? ?????? ???? h ?(n).
• ???? ?????? ?? ???? ?????? ?? ?? ????? ?? ?? ???

51
Tree Search

• Tree-Search(x, k)
• 1. if x NIL or k keyx
• 2. then return x
• 3. if k lt keyx
• 4. then return Tree-Search(leftx, k)
• 5. else return Tree-Search(rightx, k)

Running time O(h) Aside tail-recursion
52
?????? ?????? Iterative

• Iterative-Tree-Search(x, k)
• 1. while x ? NIL and k ? keyx
• 2. do if k lt keyx
• 3. then x ? leftx
• 4. else x ? rightx
• 5. return x

Search 24
56
?????? ????? ?? ???? ??????? ?????? ?? ???
??????? ???? ??? ????? ???? ??? ??????? ?????? ?
???? ?? ???
53
????? Min , Max

• ??????BST ????? ?? ???
• MIN ?? ??? ?? ???? ??? ???? ?? ????
• MAX ?? ??? ???? ???? ??? ???? ?? ????.

• Tree-Minimum(x) Tree-Maximum(x)
• 1. while leftx ? NIL 1. while
  rightx ? NIL
• 2. do x ? leftx 2. do x ?
  rightx
• 3. return x 3. return x

????? ??? ?????? ???? ??? ?
54
Predecessor Successor

• Successor (x) ??? ??? ?? x ???? ?? ????? ?? ??
  ????? x? ?????? ??? ?? ???? ??????? ?????? ?? x ?
  ?????? ???
• ???????? ???? ????
• ???? ?? ????? ?????? inorder? ???????? ??? ?? x?
  ???? ?? ???
• Predecessor(x) ??? ??? ?? x ???? ?? ????? ?? ??
  ???? x? ?????? ??? ?? ???? ??????? ?????? ??x ?
  ?????? ???
• ???????? ???? ?????
• ???? ?? ????? ?????? inorder? ???????? ??? ?? x?
  ???? ?? ???

55
??? ?? Successor

• Tree-Successor(x)
• if rightx ? NIL
• 2. then return Tree-Minimum(rightx)
• 3. y ? px
• 4. while y ? NIL and x righty
• 5. do x ? y
• 6. y ? py
• 7. return y

Successor (28)
y
x
Code for predecessor is symmetric. Running time
O(h)
56
?????? ?? ??? ?? ????

• Tree-Insert(T, z)
• y ? NIL
• x ? rootT
• while x ? NIL
• do y ? x
• if keyz lt keyx
• then x ? leftx
• else x ? rightx
• pz ? y
• if y NIL
• then roott ? z
• else if keyz lt keyy
• then lefty ? z
• else righty ? z

• ??? ?????? ???? ???? ????? ??? ?? ????? BST ???
  ???
• ???? ?????? 50 ?? ???? ???

50
57
?????? ?????? ???

• Tree-Insert(T, z)
• y ? NIL
• x ? rootT
• while x ? NIL
• do y ? x
• if keyz lt keyx
• then x ? leftx
• else x ? rightx
• pz ? y
• if y NIL
• then roott ? z
• else if keyz lt keyy
• then lefty ? z
• else righty ? z

• Initialization O(1)
• While loop in lines 3-7 searches for place to
  insert z, maintaining parent y.O(h) time.
• Lines 8-13 insert the value O(1)
• ? TOTAL O(h) time to insert a node.

58
Tree-Delete (T, x)

• if x has no children then remove x ?
  case 0
• if x has one child then point px to child ?
  case 1
• if x has two children (subtrees) then ?
  case 2
• swap x with its successor
• perform case 0 or case 1

Delete 26
Delete 28
56
56
26
200
28
28
18
18
27
27
59
??? ?? Successor

• Tree-Successor(x)
• if rightx ? NIL
• 2. then return Tree-Minimum(rightx)
• 3. y ? px
• 4. while y ? NIL and x righty
• 5. do x ? y
• 6. y ? py
• 7. return y

Successor (28)
y
x
Code for predecessor is symmetric. Running time
O(h)
60
??? ?? ???

• Tree-Delete(T, z)
• / Determine which node to splice out either z
  or zs successor. /
• if leftz NIL or rightz NIL
• then y ? z
• else y ? Tree-Successorz
• / Set x to a non-NIL child of x, or to NIL if y
  has no children. /
• if lefty ? NIL
• then x ? lefty
• else x ? righty
• / y is removed from the tree by manipulating
  pointers of py and x /
• if x ? NIL
• then px ? py
• / Continued on next slide /

61
??? ?? ??? ?? ???

• Tree-Delete(T, z) (Contd. from previous slide)
• if py NIL
• then rootT ? x
• else if y leftpi
• then leftpy ? x
• else rightpy ? x
• / If zs successor was spliced out, copy its
  data into z /
• if y ? z
• then keyz ? keyy
• copy ys satellite data into z.
• return y

62
????? ???????? ???

• ?? ???? ???? ????? ???????? ?? ???? 0 ?? 1 ?????
  ?? ???? ????
• ??? x ?? ????? ????? ????? ??? ???? ??? ????????
  ??? ?????? ??? ???? ????? ???. ??? ??? ????? ???
  ?? ?????. ??? ????? ???? ????? ????? ???? 0 ? ???
  ?????? ???? ???? 1 ????? ?? ????
• ?? ??? ???????? ?? ???? ?? ??? ??? ??? ???????
  ???. ???? ??? ????? ??? ??? ??? ?? ?????? ??? ?
  ??? ????? ?????? ?????? ??? ?? ???? ????? ????
  ???????? ???? ????.

63
?????

• ??? ?? ??? ???? ???? ???? ?????? BST ????? ???
  ???. ????? ????? ?? ?? ?? ??????? ????? ? ??????
  ???? ?????? ????
• ??????? ??? ?? ?????? ??? ???????? ?? ?????? ??
  ???? ?????? ???? ???? ???? ?
• Sort (A)
• for i ? 1 to n
• do tree-insert(Ai)
• inorder-tree-walk(root)