Trees

1
Trees
2
Introduction to Trees

• ?????? (tree) ???????????????????????????????
  ????????????????????????????????
  ?????????????????? ????????????????????? ???????
• ??????(tree) ??????????????????????????????(connec
  ted undirected graph) ????????????
• Theorem ?????????????????????????????????????????
  ????????(???)???
• ???(forest)???????????????????????????(???)???????
  
• ????????(leaf) ??????????????? ??????????????
  pendant ?????????????(isolated)
  ????????(internal)???????????????????????
  (?????????????????????? ___ )

2
3
???????????????????????????
4
Forest
5
Basic Definition

• ?????????? ?????????????????
• ?????????????????????
• ?????????????????(bond) ????????????

6
Rooted Trees

• ??????????????(rooted tree) ???
  ?????????????????????????????????????????(root)???
  ?????????????????????????????? ?????????????????
  ????????????????????????? ????????????????????????
  ?????????????(unique) ????????????????????? ?
  ???????????????
• ??????????????????????????????? n ????
  ?????????????????????????????????? n
  ????????????????

????????????????? ?????????????????????
root
root
?????????????????????? ??????????????????????????
7
Rooted Trees

• Example

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

• ???/??? (Parent)
• ??? T ?????????????????? ??? v ????????? T
  ???????????? ???????/????????? v ??? ??? u
  ???????????????????? u ????????? v ??? u
  ??? v ????????? ??? u ?????????????????? v
• ??? (Child)
• ????? u ???????/?????? v ???????????? v
  ????????? u
• ?????????? (Siblings)
• ??? 2 ?????????????? 2 ??? ????????/??????????
  ????????????????

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

• ????????? (Ancestor)
• ????????????????? ? ?????????????????????
  ???????????????????????????????????????????
• ??????? (Descendant)
• ????????????? v ??? ?????? ? ???????? v
  ?????????????
• ?? (Leaf)
• ????? ? ??????????? ??????????
• ???????? (Internal vertex/nodes)
• ????????????????? ??????????????????????????????
  ?? ??????????????????????????

10
Rooted-Tree Terminology Example
11
Rooted-Tree Terminology Example
ancestors of h and i
12
Rooted-Tree Terminology Exercise
c

• ??????? parent,children, siblings,ancestors,
  ??? descendants ??????? f

o
n
h
r
g
k, l
d
a, c
m
b
root
l, k, q
a
c
g
e
q
i
f
l
j
k
p
13
m-ary trees

• ?????????????? ?????????? m ???(m-ary tree)
  ???????????????(children)?????????? m ??? ???
  ?????????????? m ??????????(full m-ary tree)
  ??????????????????????????? m ???
• ?????? 2-ary(???????????? 2 ???)
  ??????????????????????(binary tree)
• ????? ?????????????????? binary tree
  ?????????????????(????????????) ??????? ___,
  ???????????????? ___

3
2
14
?????????????????? m ???????????
Full 3ary tree
Full 2ary tree
Full 5ary tree
Not full 3ary tree
15
Some Tree Theorems

• ??????? 1 ?????????????????? n ???
  ?????????????(????) e n-1 ????

16
Some Tree Theorems

• ??????? 2 ?????? m ??????????(full m-ary tree)
  ?????????????????? i ??? ???????????????????
  nmi1 ??? ?????????? ?(m-1)i1 ???

17
Some Tree Theorems

• ??????? 3 ????????? m ?????????????????? n
  ????????????????????? i ?????????????????? ?
  ??? ????????
• i (n-1)/m ??? ? (m-1)n1/m
• n mi 1 ??? ??? ? (m-1)i 1
• n (m? - 1)/(m-1) ??? i (? -1)/(m-1)

18
????????

• ??????????????????????????????????????????????????
  ????????????????? ? ??????????????????????????????
  ?????????????????????????????? ? ??? 4 ??
  ??????????????????????????????????????????????????
  ??????????????????????? ?????????????????????????
  ??????????????????? ?????????????????????????????
  ???????????? ????????? 100 ?????????????????????
  ?????????????????????

19
????????

• ?????????????????????????????????????????? 4 ???
  ?????????????? ??????????????????????????????????
  ??????????????????????????? ??? i
  ???????????????? ??? n ?????????????????
  ???????? m 4 ???????? ????????
• n (m? -1)/m-1
• (4 ? 100)-1/(4-1) 133
• ??????????????????????????????? 133 ??
• ??? i (? -1)/(m-1)
• (100-1)/(4-1) 33
• ?????????????????????????????????????? 33 ??

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Some More Tree Theorems

• ????? ?????(level) ??????????????????????????????
  ???????????(root)????????????

level 2
21
Some More Tree Theorems

• ???????(height) ????????? ????????????????????????
  ??

??????????? (??? a) 0 ??????????? b, j,
k 1 ??????????? c, e, f, l
2 ??????????? d, g, i, m, n 3 ???????????
h 4 ??????? ???????????????? T 4
22
Some More Tree Theorems

• ?????? m ???(m-ary tree) ???????? ???????????? h
  ???????????????????(balanced) ????????????????????
  ?????? h ???? h-1
• ??????? ??????????????????????? mh ?? ????????
  m ???(m-ary tree) ???????????? h
• ??????????? ?????? m-ary ?????????? ?
  ??????????? h?logm?? ?????????????????????(
  full m-ary) ??????????????? h?logm??

23
Applications of Trees Binary Search Tree Format

• ???????????????????????????????????????
  ????????????????????????????????????????? x ???,
• ??????????????????????????????????(left
  subtree)??? x ??????????????? x
• ?????????????????????????????????(right
  subtree)??? x ?????????????? x

????????
7
3
12
1
5
9
15
0
2
8
11
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Applications of Trees Codes

• Codes That Can Be Decoded
• Fixed-length codes
• Every character is encoded using the same number
  of bits.
• To determine the boundaries between characters,
  we form groups of w bits, where w is the length
  of a character.
• Examples
• ASCII
• Prefix codes
• No character is the prefix of another character.
• Examples
• Huffman codes

25
Why Prefix Codes ?

• ?????????????????????????????? a prefix codea
  01 m 10 n 111 o 0 r 11 s
  1 t 0011
• ???????????????????????? You are a
  star.?????????????????????????????? star ???
  1 0011 01 11
• ??????????????????????????????????????????(decodes
  )????????????????????????? ????????????????????
  100110111 10 0 11 0 111 moron

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Representing a Prefix-Code Dictionary

• Our example space 000 A 0010 E
  0011 s 010 c 0110 g 0111 h 1000
  i 1001 l 1010 t 1011 e 110 n
  111

1
0
0
0
1
1
0
0
0
0
1
1
1
1
s
n
e
spc
1
0
0
0
0
1
1
1
g
t
c
h
i
A
E
l
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Huffman code

• ??????????????????????????????????????????????????
  ???????????prefix code
• ?????????????????????? Huffman code
  ??????????????????????????
• 1. ???????????????????????????????????????????????
  ?????????????????????????
• 2. ??????????????????? 2 ????????????????
  ??????????????????????????????????
• 3. ??????????????????????????
• 4. ????????????2 ??? 3 ???????????????????????????
  ?????????????????????????????
• 5. ???????????? ????????? 0 ??????????????????????
  ??????? 1 ????????????????(???????????????????????
  ????????) ??????????? Huffman tree

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Applications of Trees

• Q Use Huffman coding to encode the following
  five symbols with given frequencies
• A 45 , B 13 ,C 12 , D 16 , E 9, F 5

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Huffmans Algorithm
30
Huffmans Algorithm
31
Huffmans Algorithm
32
Huffmans Algorithm
33
Huffmans Algorithm
34
Huffmans Algorithm
35
Huffmans Algorithm
36
Huffmans Algorithm
37
Huffmans Algorithm
38
Tree Traversal

• Is a procedure that systematically visits every
  vertex of an ordered rooted tree.
• Visiting a vertex processing the data at the
  vertex.
• ?????????????????????????????(Traversal
  algorithms)
• Pre-order traversal NLR (node,left,right)
• In-order traversal LNR (left,node,right)
• Post-order traversal LRN (left,right, node)

39
Tree Traversal

• Let T be an ordered rooted tree with root r.
  Suppose that T1, T2, , Tn are the subtrees at r
  from left to right in T.

r
55
T2
T1
24
94
49
60
3
88
46
58
52
91
76
45
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Preorder Traversal

• Begins by visiting r.
• Next traverses T1 in preorder, then T2 in
  preorder, , then Tn in preorder.
• Ends after Tn has been traversed.

41
Preorder Traversal

• Visit root, visit subtrees left to right.

42
Preorder Traversal
43
Inorder Traversal

• Begins by traversing T1 in inorder
• Next visit r, then traverses T2 in inorder, ,
  then Tn in inorder.
• Ends after Tn has been traversed.

44
Inorder Traversal

• Visit leftmost subtree,visit root, visit other
  subtrees left to right.

45
Inorder Traversal
46
Postorder Traversal

• Begins by traversing T1 in postorder
• Next traverses T2 in postorder,then T3 in
  postorder,...,then
• Tn in postorder
• End by visiting r

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Postorder Traversal

• Visit subtree left to right,visit root.

48
Postorder Traversal
49
Postorder Traversal

• Applications compute space used by files in a
  directory and its subdirectories.

50
Tree Traversal Shortcut

• Preorder list is obtained by
• listing each vertex the first time this curve
  passes it.
• Result a, b, d, h, e, i, j, c, f, g, k.
• Inorderlist is obtained by
• listing a leaf the first time the curve passes it
  and
• listing each internal vertex the second time the
  curve passes it.
• Result h, d, b, i, e, j, a, f, c, k, g.
• Postorderlist is obtained by
• listing a vertex the last time it is passes on
  the way back up to its parent.
• Result h, d, i, j, e, b, f, k, g, c, a.

51
Representing Arithmetic Expressions

• Complicated arithmetic expressions can be
  represented by an ordered rooted tree
• Internal vertices represent operators
• Leaves represent operands
• Build the tree bottom-up
• Construct smaller subtrees
• Incorporate the smaller subtrees as part of
  larger subtrees

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Example

• (xy)2 (x-3)/(y2)

53
Infix Notation

• Traverse in inorder adding parentheses for each
  operation

x
y
2


x

3
y

2
?
/
54
Prefix Notation(Polish Notation)

• Traverse in preorder

x
y
2


x

3
y

2
?
/
55
Evaluating Prefix Notation

• In an prefix expression, a binary operator
  precedes its two operands
• The expression is evaluated right-left
• Look for the first operator from the right
• Evaluate the operator with the two operands
  immediately to its right

56
Example
/ 2 2 2 / 3 2 1 0
/ 2 2 2 / 3 2 1
/ 2 2 2 / 1 1
/ 2 2 2 1
/ 4 2 1
2 1
3
57
Postfix Notation(Reverse Polish)

• Traverse in postorder

?
/



2
y
y
x
x
3
2
x
y
2


x

3
y

2
?
/
58
Evaluating Postfix Notation

• In an postfix expression, a binary operator
  follows its two operands
• The expression is evaluated left-right
• Look for the first operator from the left
• Evaluate the operator with the two operands
  immediately to its left

59
Example
2 2 2 / 3 2 1 0 /
4 2 / 3 2 1 0 /
2 3 2 1 0 /
2 1 1 0 /
2 1 1 /
2 1
3
60
Tree Traversal
-
prefix expression
5

-
(
(
(
3
4
)
2
8


(
(
)
)
)
7

)

7
infix expression

5

(
3
4
)
2
7

5

-
(
8

(
)
)
)
)
(
(

2
postfix expression

8

3
4
2
7

5

-
8

4
3