???????????????????????? (Tree)

1
????????????????????????(Tree)
2
Tree

• ??????? ??????????????????????????????????????????
  ?????????? ???????????????????????????????????????
  ???? ?????????????????????????????????????? Root
  Node ??????????????????????????????
  ????????????????????????????????????????
  ???????????? Subtree

3
???????????????????????

• ???????????????????????? ?????????????????
• 1. ????????????????? ?????????? (Root node) ,
  R
• 2. ??????????????????????????????? n ?????
  ????????????????????????? ?????????? ???? ?????
  T1 , T2 ,..Tn (n gt0) ???????????????????????????
  ???? ???????????????????? (Subtree)

4
????????????
?????? R ???????????????????? A,B,C,D A
???????????????????? E,F,G F ????????????????????
J B ???????????????????? H ??? I
5
??????????????????????

6
???????????? (Level)

• ????????????????? ? ??????????????????????????????
  ?????????????????????????????????????
  ??????????????????????????????????????????? 1
  ?????????????????????????????? ????????? 1 ?????
  ????????????????? ? ??????????????????????????????
  ????????????????? F ?????????????? 4 ???????

7
???????????? (Level Degree)

• ???????????? ??? ??????????????????????????
  ?????? ???? X ??????? 1 ???? A ??????? 2 ????????
  H ??????? 3 ???? B ??????? 1 ??????? E ??????? 0
  ???????

8
????????????? (Leaf Node)

• ????????????? ????????????????????????? 0
  ???????? C, D, E, J, F ??? G ?????????????????????
  ??????? 0 ???????? ????????? ???? interior node
  ???? branch node

9
Immediate Successor ???? SON ??????? i

• Immediate Successor ????????????????????????????
  i ???????????????????????? i ????????? ???? SON
  ??????? H ??????? E, I, ??? F

10
Immediate Predecessor ???? father ??????? i

• Immediate Predecessor ????????????????????????????
  ??? i ????????? ???? FATHER ??????? J ??????? I,
  FATHER ??????? I ??????? H ???????

11
??????????????? (Tree Structure)
R
Root Nodes
Nodes X ???? Immediate Predecessor ??? Y ???? X
???? Father (Parent) ??? Y
B
A
X
D
C
H
Y
Nodes Y ???? Immediate Successor ??? X ???? Y
???? son (Child) ??? X
I
E
G
Leaf Nodes
12
??????????????? (Tree Structure)
Level ????????????????????????????????????????????
?????? Root Node ????????????????????????????????
??????????? 1 ?????
13
??????????????? (Tree Structure)
Degree ??????????????? Subtree ??????????? ???? A
?? Degree 2, X ?? Degree 1
14
??????????????? (Tree Structure)
Leaf Node ???????????????? Degree 0 ???? C, D,
E, I ,G ???????? ????? Degree ltgt 0 ????????
Branch Node ???? Interior Node
15
??????????????? (Tree Structure)
Nodes X ???? Immediate Predecessor ??? Y ???? X
???? Father (Parent) ??? Y
Nodes Y ???? Immediate Successor ??? X ???? Y
???? son (Child) ??? X
Predecessor ??????? ??? Node ??????????????? Succe
ssor ??????? ??? Node ???????????? ???? R, B,
H ???? Predecessor ??? E, I, I ????
Successor ??? H
16
??????????????? (Binary Tree)

• ???????????????? rooted binary tree ????????????
  ???????????????????????????????????? 2
  ???????????????????????? ???????????????????????
  (?????????????????????????????)
  ???????????????????? (left subtree)
  ??????????????????? (right subtree)
  ????????????? ????????? ??????????????????
  ??????????????????????????????????????????????????
  ???

17
A
C
S
V
B
C
G
F
D
D
E
F
G
(?)
X
(?)
(?)
??? (?), (?) ??? (?) ????????????????
18
(?)
(?)
(?)
??? (?), (?) ??? (?)??????????????????
19
?????????????????????? (Complete Binary Tree)

• ?????????????????????? ??????????????????????????
  ?????????????????????????????? (??????????????????
  ????? left son ??? right son ) ????????? (leaf
  nodes) ?????????????????????? n ??? (?)
  ??????????????????????????????? 3 ?????

D
R
K
M
W
O
U
(?)
20

• ????????????????????????????????????????????? n
  ?????????????????????? 2n-1

W
V
D
N
S
A
Q
?????? ???????????????? 23-1 7 ????
21
???????????????????????????????

• ???????????????????????? 2 ??? ???
• 1. ????????????????????????
• 2. ???????????????????????????? ?????????????
  ???????????
• (sequential)

22
????????????????????????

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

DATA
DATA
????
LLINK RLINK
LSON RSON
???????????????????????????????
LLINK ???? LSON ??????????????????????????????????
????? ???? RLINK ???? RSON ??????????????????????
????????????????
23
???????????????????????????? ????????????????????
????

• ?????????????????????????????????? 1
  ????????????????? ???????????????????????????????
  ????????????? complete binary tree ??????
  ?????????????????????????????????????????????
  ???????????? 1 ????? 2 ...????????? ?????????? k
  ?????????????????????????????????????????
  ?????????????????????? 1 ????
  ??????????????????????????????????????????????????
  ???????????? ???????????????????????????????

24
1
A
3
2
B
X
4
6
7
5
C
D
13
11
8
14
15
9
10
12
E
??????????????????????????????????????????????????
?
2
4
5
7
8
9
10
11
12
13
14
15
1
3
6
B
X
C
D
-
-
E
-
-
-
A
-
-
-
-
????????????????????????????? ????????????????
??????????? x ??????????????????????? 2x
?????????????????????????? 2x1
25
?????????? Tree ??????? Binary Tree
?????????????????(ordinary) ?????????????????????
???????????????????????????????
????????????????????? binary tree
?????????????????????????????????????????????
??????????????? 1. ?????????????????????????????
???????????????? 2. ?????????????????????????????
???? 1 ???????????????????????????????????????????
?????????? 3. ????????? 1 ??? 2
???????????? 4. ??????????????????? ?????? 45
????
26
?????????? Tree ??????? Binary Tree
??????? 1
??????? 4
??????? 2
??????? 3
27
?????????? Tree ??????? Binary Tree
A
???????? ????????????????????????????????????

?????
B
F
A
B
F
C
K
D
G
J
C
K
D
G
J
E
H
L
I
E
H
A
L
I
B
F
??????? 1-3
C
G
D
J
K
???????
E
L
H
I
28
????????????? (Tree Traversal)
Tree Traversal ???????????????????????????????????
??????????????????????????????? ????
????????? ??????????? 3 ???? (??????????) 1.
Pre-Order Traversal (RTLTR) 2. In-Order
Traversal (TLRTR) 3. Post-Order Traversal
(TLTRR)
29
??????????? Binary Tree
??????????????? ??????? ????????????????????????
??????????????????????????? ?????????????????
??????????????? ??????????????????? 3 ???? 1.
Inorder traversal ???? Symmetric order
?????????????????????????????????????????????????
??????????????????????????????????????????????????
??????????? (Left/ Root/Right) 2. Preorder
traversal ????????????????????????
??????????????????????????????????????????
????????????????????????????????????????????
(Root/ Left/ Right) 3. Postorder traversal ????
Endorder ???????????????????????????????????????
?????????? ???????????????????????????????????
?????????? ???????????????? (Left/ Right/ Root)
30
??????????? Binary Tree
A
1. ????????????? (Left/ Root/Right)
??????????? BAC 2. ?????????????(Root/ Left/
Right) ??????????? ABC 3. ?????????????(Left/
Right/ Root) ??????????? BCA 1.
????????????? (Left/ Root/Right) ???????????
DB A EG C HFI 2. ?????????????(Root/ Left/
Right) ??????????? A BD CEG FHI 3.
?????????????(Left/ Right/ Root) ???????????
DB GE HIF C A
B
C
A
B
C
D
E
F
G
H
I
31
??????????????????? (Binary Tree Traversal)
start
stop
??? Pre-Order Traversal
Y
Result
R
A
C
D
X
Y
G
32
??????????????????? (Binary Tree Traversal)
start
stop
??? In-Order Traversal
Y
Result
C
A
D
R
G
Y
X
33
??????????????????? (Binary Tree Traversal)
start
stop
??? Post-Order Traversal
Y
Result
C
D
A
G
Y
X
R
34
A
B
C
D
E
F
G
H
I
Pre-order ABDGCEHIF In-order
DGBAHEICF Post- order GDBHIEFCA
35
EX1

-
A

D
B
C
???????????????????? ??? 1.Pre-order
2.In-order 3.Post-order
36
Expression Tree
?????????????????????? ????????? operands,
??????????????????????, ????????????? operators
?????????? Expression tree ??? (a b c)
((d e f ) g)
37
Expression Tree
????????????????????????????????(operand)
??????????????(operators) ??????????????????????
??????????????????????????(leave node)
????????????????????????????????
?????????????????????????????????????????(unary
operator) ????????????????????????????????
??????????? ??????????????????????????????????????
??????????????????????????????????????????
?????? - log() cos()
?????????????????????? ???????????????????????????
????? ?????? ??????????????????
??????????????? ???????? ????????????
??????????????????????? ????????? X Y Z


Z
X
Y
38
Expression Tree

• ???????? ??????????????????????????????????? (X
  ((Y / R) D))
• ????????
• ????????????? ?????????????????? X-Y/RD
  ????????????????????????
• ????????????? ?????????????????? -X/YRD
  ????????????????????????
• ????????????? ????????????? ????? XYR/D-
  ????????????????????????

-
X

/
D
Y
R
39
???????? Expression Tree

• ?????????????????????????????? ???????????????????
  operand ??????????????? tree ????????????? push
  ??????? stack ??????????????????? operator ???
  pop ??? stack 2 ????? ????????? trees T1 ??? T2
  (T1 ?????????) ???????????????????? tree
  ???????????? (root) ??????? operator ????? left
  ??? right children ???? T2 ??? T1 ????????
  ????????????? tree ????????????? stack

40

• ????????? ??? a b c d e

1
2
3
41

• ????????? ??? a b c d e

4
5
42

• ????????? ??? a b c d e

6
7
43
?????????

• 1. ??????????????????????? ???????????????????????
  ???????????????? ??? Preorder ,Inorder ,
  Postorder

A
H
B
J
F
G
I
D
K
M
2. ??????? Expression Tree ?????????????????
2.1 (A - 2 (B C) D E) F 2.2 A
(B C) D ( E F )
44
?????????

• 3. ?????????????????????????

A
H
B
J
C
F
G
I
D
E
K
M