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Chapter 6. Introduction to Trees
[INA240] Data Structures and Practice
Youn-Hee Han
http://link.kut.ac.kr
1. Basic Tree Concepts
Logical structures
Chap. 3~5
Chap. 6~10
Chap. 11
Linear list
Tree
Graph
Linear structures
Non-linear structures
1. Basic Tree Concepts
선형 자료 구조

리스트, 스택, 큐가 데이터 집합을 한 줄로 늘어 세운 선형 자료구조
비선형 자료 구조

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트리(Tree)는 데이터 집합을 여러 갈래로 나누어 세운 비 선형 구조
비선형 구조 고안 동기
 효율성

3
삽입, 삭제, 검색에 대해서 선형 구조보다 나은 시간적 효율을 보임
Data Structure
1. Basic Tree Concepts
Tree consists of
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A finite set of nodes (vertices)
A finite set of branches (edges, arcs) that connects the nodes
 Degree of a node: # of branches
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In-degree: # of branch toward the node (# of upward branch)
Out-degree: # of branch away from the node (# of downward branch)
Every non-empty tree has a root node whose in-degree is zero.
In-degree of all other nodes except the root node is one.
Data Structure
1. Basic Tree Concepts
Definitions
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트리의 구성 요소에 해당하는 A, B, C, ..., F를 노드 (Node)라 함.
A node is child of its predecessor
 B의 바로 아래 있는 E, F를 B의 자식노드 (Child Node)라 함.

A node is parent of its successor nodes
 B는 E, F의 부모노드 (Parent Node)
 A는 B, C, D의 부모노드 (Parent Node)임.

어떤 노드의 하위에 있는 노드를 후손노드 (Descendant Node)라 함.
 B, E, F, C, D는 A의 후손노드임.

주어진 노드의 상위에 있는 노드들을 조상노드 (Ancestor Node)라 함.
 B, A는 F의 조상노드임.

Path
 a sequence of nodes in which each node
is adjacent to the next one
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Data Structure
1. Basic Tree Concepts
Definitions

자매노드 (Sibling Node): nodes with the same parent
 같은 부모 아래 자식 사이에 서로를 자매노드 (Sibling Node)라 함.
 B, C, D는 서로 자매노드이며, E, F도 서로 자매노드임.
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부모가 없는 노드 즉, 원조격인 노드를 루트 노드 (Root Node)라 함.
C, D, E, F처럼 자식이 없는 노드를 리프노드 (Leaf Node)라 함.
 리프노드를 터미널 노드 (Terminal Node)라고도 함.
 node with out-degree zero

리프노드 및 루트노드를 제외한 모든 노드를 내부노드 (Internal
Node)라 함
 B
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Data Structure
1. Basic Tree Concepts
Definitions

트리의 레벨(Level)
 distance from the root
 루트노드를 레벨 0으로 하고 아래로 내려오면서 증가

트리의 높이(Height or Depth)
 level of the leaf in the longest path from the root + 1
 (트리의 최대 레벨 수 + 1) = 트리의 높이(Height)
 루트만 있는 트리의 높이는 1
 비어있는 트리 (empty tree)의 높이는 0
(=4)
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Data Structure
1. Basic Tree Concepts
Definitions
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서브트리 (Subtree)
주어진 트리의 부분집합을 이루는 트리를 서브트리 (Subtree)라 함.
 서브트리는 임의의 노드와 그 노드에 달린 모든 후손노드 (Descendant
Node) 를 합한 것임.
 임의의 트리에는 여러 개의 서브트리가 존재할 수 있음.
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B, E, F가 하나의 서브트리  Subtree B
C 자체로서도 하나의 서브트리  Subtree C
Data Structure
1. Basic Tree Concepts
Definitions

서브트리 (Subtree)
Subtree B can be
divided into two
subtrees, C and D
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Data Structure
1. Basic Tree Concepts
Recursive Definitions of Tree

A tree is a set of nodes that either:
 1. is Empty, or
 2. Has a designated node, called the the root, from which hierarchically
descend zero or more subtrees, which are also trees.
User Representations
General tree
Indented list
Parenthetical listing
A (B (C D) E F (G H I) )
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Data Structure
1. Basic Tree Concepts
트리의 높이(Height)에 대한 다른 정의

1 + 루트노드에서 가장 먼 리프노드까지의 연결 링크(Link) 개수
트리의 높이(Height)에 대한 재귀적 정의

트리의 높이
= 1 + Max{왼쪽 서브트리의 높이, 오른쪽 서브트리의 높이}
Height 3
Height 1
Height 4
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Data Structure
2. Binary Tree
General Tree
 임의의 임의의 노드가 둘 이상의 자식노드를 가지는 트리를 일반트리
(General Tree)라 함.
Binary Tree

임의의 노드가 최대 두 개까지의 자식노드를 가질 수 있는 트리를 이진트리
(Binary Tree)라 함.
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a tree in which no node can have more than two subtrees
the child nodes are called left and right
Left/right subtrees are also binary trees
Data Structure
2. Binary Tree
이진트리의 재귀적 정의 (Recursive Definition)
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1) 아무런 노드가 없는 트리이거나,
2) 가운데 노드를 중심으로 좌우로 나뉘되, 좌우 서브트리 (Subtree)
모두 이진트리다.
1)번 때문에,
아무런 노드가 없는 트리도
주어진 트리의 서브트리이며
동시에 이진트리가 됨.
1)번은 재귀호출의 Base-case에 해당
13
Data Structure
2. Binary Tree
Examples of Binary Trees
14
Data Structure
2. Binary Tree
Properties of Binary Tree

트리 내의 노드의 수를 N 이라 가정했을 때
 Maximum Height
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Ex] Given N=3 in a binary tree, what is the maximum height?
 Minimum Height
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H max  N
H min  log 2 N  1
Ex] Given N=3 in a binary tree, what is the minimum height?
트리의 높이를 H 라고 가정했을 때
 Maximum Number of Nodes
N max  2 H  1
Ex] Given H=3 in a binary tree, what is the maximum number of nodes?
 Minimum Number of Nodes
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N min  H
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Ex] Given H=3 in a binary tree, what is the minimum number of nodes?
Data Structure
2. Binary Tree
Definitions
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삼진트리 (Ternary Tree)
Data Structure
2. Binary Tree
Balance

검색에 있어서 다음 어느 Tree 가 효율이 더 좋을까?
 높이가 3인 Tree
 높이가 2인 Tree

The shorter the tree, the easier it is to locate any desired node in the
tree
Balance Factor
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H L : the height of the left subtree
H R: the height of the right subtree
Balance Factor
B  HL  HR
Balanced Binary Tree
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A tree with |B|  1
HL
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HR
Data Structure
2. Binary Tree
Complete Binary Tree (완벽 이진트리)
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높이가 h인 이진트리에서 모든 리프노드가 레벨 h-1에 있는 트리
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리프노드 위쪽의 모든 노드는 반드시 두개의 자식노드를 거느려야 함.
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시각적으로 볼 때 포화될 정도로 다 차 들어간 모습.
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A bianary tree with the maximum # of entries for its height
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A binary tree in which all leaves (vertices with zero children) are at the
same depth
H
Complete binary tree의 Height가 H 일 때  node의 갯수가 N max  2  1
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Data Structure
2. Binary Tree
Nearly Complete Binary Tree
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레벨 (L-1)까지는 완벽 이진트리
마지막 레벨 L에서 왼쪽에서 오른쪽으로 가면서 리프노드가 채워짐.
 하나라도 건너뛰고 채워지면 안됨.

Complete Binary Tree 이면 Nearly Complete Binary Tree 이다. 그러나
역은 성립 안 됨.
Complete binary tree의 노드의 수가 N 일 때  height는 H min  log 2 N  1
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Data Structure
2. Binary Tree – Binary Tree Traversal
Binary Tree Traversal
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process each node once and only once in a predetermined
sequence
Two general approach
 Depth-first traversal (depth-first search: DFS)
 Breadth-first traversal (breadth-first search: BFS, level-order)
1
1
5
2
3
4
6
depth-first traversal
20
3
2
7
4
5
6
7
breadth-first traversal
Data Structure
2. Binary Tree – Binary Tree Traversal
Depth-first traversal (Depth-first search, DFS)
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The processing proceeds along a path from the root through one child
to the most distant descendent of that first child before processing a
second child.
세가지 DFS 방법

Preorder traversal (NLR)
 Root  left subtree  right subtree
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Inorder traversal (LNR)
 Left subtree  root  right subtree
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Postorder traversal (LRN)
 Left subtree  right subtree  root
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Data Structure
2. Binary Tree – Binary Tree Traversal
Depth-first traversal

Preorder traversal (NLR) – 1/2
void preOrder(root) {
if (root == NULL) return;
printf("%s", root->data);
preOrder(root->left);
preOrder(root->right);
}
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Data Structure
2. Binary Tree – Binary Tree Traversal
Depth-first traversal

Preorder traversal (NLR) – 2/2
void preOrder(root) {
if (root == NULL) return;
printf("%s", root->data);
preOrder(root->left);
preOrder(root->right);
}
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Data Structure
2. Binary Tree – Binary Tree Traversal
Depth-first traversal

Inorder traversal (LNR)
void inOrder(root) {
if (root == NULL) return;
inOrder(root->left);
printf("%s", root->data);
inOrder(root->right);
}
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Data Structure
2. Binary Tree – Binary Tree Traversal
Depth-first traversal

Postorder traversal (LRN)
void postOrder(root) {
if (root == NULL) return;
postOrder(root->left);
postOrder(root->right);
printf("%s", root->data);
}
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Data Structure
2. Binary Tree – Binary Tree Traversal
Breadth-first traversal (=level-order traversal)
Begins at the root node and explores all the neighboring nodes
 Then for each of those nearest nodes, it explores their
unexplored neighbor nodes, and so on, until it finds the goal
 Attempts to visit the node closest to the root that it has not
already visited

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Data Structure
2. Binary Tree – Binary Tree Traversal
Breadth-first traversal (=level-order traversal)
void BForder(TreeNode *root) {
QUEUE *queue = NULL;
TreeNode *node = root;
if (node == NULL)
return;
queue = createQueue();
while(node){
process(node->data);
if(node->left) enqueue(queue, node->left);
if(node->right) enqueue(queue, node->right);
if(!emptyQueue(queue))
dequeue(queue, (void**)&node);
else
node = NULL;
}
destroyQueue(queue);
}
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Data Structure
2. Binary Tree – Tree Examples
Expression Tree: a binary tree in which
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Each leaf is an operand
Root and internal nodes are operators
Subtrees are sub-expressions with root being an operator
Traversal Methods
 Infix, Postfix, Prefix
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Data Structure
2. Binary Tree – Tree Examples
Infix Traversal in Expression Tree
void infix(TreeNode *root) {
if(root == NULL)
return;
if(root->left == NULL && root->right == NULL)
printf("%s", root->data);
else {
printf("(");
infix(root->left);
printf("%s", root->data);
infix(root->right);
printf(")");
}
}
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Data Structure
2. Binary Tree – Tree Examples
Postfix Traversal in Expression Tree
void postfix(TreeNode *root) {
if(root == NULL)
return;
postfix(root->left);
postfix(root->right);
printf("%s", root->data);
}
abc+*d+
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Data Structure
2. Binary Tree – Tree Examples
Prefix Traversal in Expression Tree
void prefix(TreeNode *root) {
if(root == NULL)
return;
printf("%s", root->data);
prefix(root->left);
prefix(root->right);
}
+*a+bcd
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Data Structure
2. Binary Tree – Tree Examples
Huffman Code

Representation of characters in computer
 7 bits/char (ASCII)
 2 bytes/char (KSC Hangul)
 2 bytes/char (UNICODE)
 Isn’t there more efficient way to store text?

Data compression based on frequency  Huffman Code
 Variable length coding
 Assign short code for characters used frequently
In a text, the frequency of the character E is 15%, and the
frequency of the character T is 12%
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Data Structure
2. Binary Tree – Tree Examples
Huffman Code

Building Huffman tree
1. Organize entire character node in a row, ordered by frequency
2. Repeat until all nodes are
connected into “One Binary Tree”
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Find two nodes with the lowest
frequencies
Merge them and make a binary
sybtree
Mark the merged frequency into
the root of the subtree
3. End of repeat
33
Data Structure
2. Binary Tree – Tree Examples
Huffman Code

Building Huffman tree
Huffman Tree
34
Data Structure
2. Binary Tree – Tree Examples
Huffman Code

Getting Huffman code from Huffman tree
 Assign code to each character according to the path from root to
the character

Properties of Huffman code
 The more frequently used a character, the shorter its code is
 No code is a prefix of other code
35
Data Structure
2. Binary Tree – Tree Examples
Huffman Code

Data Encoding for communication
 AGOODMARKET (11 char. * 7 bits = 77 bits)
  00011010001001011111000000101011011100111 (42bits)

Decoding
 00011010001001011111000000101011011100111
  AGOODMARKET
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Data Structure
2. Binary Tree – Tree Examples
Huffman Code

37
Another Example
문자
빈도
A
2
B
18
C
9
D
30
E
9
F
36
104
66
38
A = 0110
B
D
20
F
B = 00
C = 0111
D = 10
11
E
E = 010
F = 11
A
C
문자
빈도
원래 크기
압축된 크기
차이
A
2
7*2=14
4*2=8
6
B
18
7*18=126
2*18=36
90
C
9
7*9=63
4*9=36
27
D
30
7*30=210
2*30=60
150
E
9
7*9=63
3*9=27
36
F
36
7*36=252
2*36=72
180
계
104
728
240
488
Data Structure
3. General Tree
General Tree

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a tree in which each node can have an unlimited out-degree
Data Structure
3. General Tree
Insertion in General Tree

FIFO insertion
 For a given parent node, insert the new node at the end of sibling
list

LIFO insertion
 For a given parent node, insert the new node at the beginning of
sibling list
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Data Structure
3. General Tree
Insertion in General Tree

Key-sequenced insertion
 places the new node in key sequence among the sibling nodes
 Most common of the insertion rules in general tree
 Similar to the insertion rule in a general ordered linked list
40
Data Structure
3. General Tree
Deletion in General Tree

Deletion of only leaf node
 A node cannot be deleted if it has any children

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Other rules for deletion…
Data Structure
3. General Tree
General Tree to Binary Tree

A general tree can be represented by a binary tree

변환 이유
 Ex] 일반트리에서는 자식노드의 수가 몇 개가 있는지 예측 불가

변환 방법
 부모노드 (Parent Node)는 무조건 첫 자식노드 (First Child Node)를 가리킴
 첫 자식노드로(First Child Node)부터 일렬로 자매 노드 (Sibling Node)들을
연결
42
Data Structure
3. General Tree
General Tree to Binary Tree
43
Data Structure