1. Chapter 6. Introduction to Trees
Internet Computing KUT
Youn-Hee Han

2. 1. Basic Tree Concepts Logical structures Chap. 3~5 Chap. 6~10
Linear list
Tree
Graph
Linear structures
Non-linear structures

3. 1. Basic Tree Concepts 선형 자료 구조 비선형 자료 구조
리스트, 스택, 큐가 데이터 집합을 한 줄로 늘어 세운 선형 자료구조
비선형 자료 구조
트리(Tree)는 데이터 집합을 여러 갈래로 나누어 세운 비 선형 구조
비선형 구조 고안 동기
효율성
삽입, 삭제, 검색에 대해서 선형 구조보다 나은 시간적 효율을 보임
Data Structure

4. 1. Basic Tree Concepts Tree consists of
A finite set of nodes (vertices)
A finite set of branches (edges, arcs) that connects the nodes
Degree of a node: # of branches
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 is 1
Data Structure

5. 1. Basic Tree Concepts Definitions
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.
주어진 노드의 상위에 있는 노드들을 조상노드 (Ancestor Node)라 함.
B, A는 F의 조상노드임.
어떤 노드의 하위에 있는 노드를 후손노드 (Descendant Node)라 함.
B, E, F, C, D는 A의 후손노드임.
Path
a sequence of nodes in which each node is adjacent to the next one
Data Structure

6. 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.
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
Data Structure

7. 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)
Data Structure

8. 1. Basic Tree Concepts Definitions
서브트리 (Subtree): any connected structure below the root
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.
B, E, F가 하나의 서브트리  Subtree B
C 자체로서도 하나의 서브트리  Subtree C
Subtree B can be divided into two subtrees, C and D
Data Structure

9. 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
Parenthetical listing
Indented list
A (B (C D) E F (G H I) )
Data Structure

10. 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 1
Height 3
Height 4
Data Structure

11. 2. Binary Tree Definitions
임의의 노드가 둘 이상의 자식노드를 가지는 트리를 일반트리 (General Tree)라 함.
임의의 노드가 최대 두 개까지의 자식노드를 가질 수 있는 트리를 이진트리 (Binary Tree)라 함.
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

12. 2. Binary Tree 이진트리의 재귀적 정의 (Recursive Definition)
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
Data Structure

13. 2. Binary Tree
Examples of Binary Trees
Data Structure

14. 2. Binary Tree Properties of Binary Tree 가정(Assumption): 트리 내의 노드의 수:
Maximum Hight
Ex] Given N=3 in a binary tree, what is the maximum hight?
Minimum Hight
Ex] Given N=3 in a binary tree, what is the minimum hight?
가정(Assumption): 트리의 높이:
Maximum Nodes
Ex] Given H=3 in a binary tree, what is the maximum numbers of nodes?
Minimum Nodes
Ex] Given H=3 in a binary tree, what is the minimum numbers of nodes?
Data Structure

15. 2. Binary Tree
Definitions
삼진트리 (Ternary Tree)
Data Structure

16. 2. Binary Tree Balance Balance Factor Balanced Binary Tree
The shorter the tree, the easier it is to locate any desired node in the tree
Balance Factor
: the height of the left subtree
: the heighr of the right subtree
Balanced Binary Tree
A tree with |B|  1 HL HR
Data Structure

17. 2. Binary Tree Complete Binary Tree (완벽 이진트리)
높이가 h인 이진트리에서 모든 리프노드가 레벨 h-1에 있는 트리
리프노드 위쪽의 모든 노드는 반드시 두개의 자식노드를 거느려야 함.
시각적으로 볼 때 포화될 정도로 다 차 들어간 모습.
A bianary tree with the maximum # of entries for its height
A binary tree in which all leaves (vertices with zero children) are at the same depth
Complete binary tree의 Height가 H 일 때  node의 갯수가
Data Structure

18. 2. Binary Tree Nearly Complete Binary Tree 레벨 (L-1)까지는 완벽 이진트리
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
Data Structure

19. 2. Binary Tree – Binary Tree Traversal
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 4 6 2 5 7 3 1 5 6 2 3 7 4
depth-first traversal
breadth-first traversal
Data Structure

20. 2. Binary Tree – Binary Tree Traversal
Depth-first traversal
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.
Preorder traversal (NLR)
Root  left subtree  right subtree
Inorder traversal (LNR)
Left subtree  root  right subtree
Postorder traversal (LRN)
Left subtree  right subtree  root
Data Structure

21. 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); }
Data Structure

22. 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); }
Data Structure

23. 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); }
Data Structure

24. 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); }
Data Structure

25. 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
Data Structure

26. 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);
Data Structure

27. 2. Binary Tree – Tree Examples
Expression Tree: a binary tree in which
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
Data Structure

28. 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);
infix(root->right);
printf(")"); }
Data Structure

29. 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+
Data Structure

30. 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
Data Structure

31. 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%
Data Structure

32. 2. Binary Tree – Tree Examples
Huffman Code
Building Huffman tree
Organize entire character node in a row, ordered by frequency
Repeat until all nodes are connected into “One Binary Tree”
Find two nodes with the lowest frequencies
Merge them and make a binary sybtree
Mark the merged frequency into the root of the subtree
End of repeat
Data Structure

33. 2. Binary Tree – Tree Examples
Huffman Code
Building Huffman tree
Huffman Tree
Data Structure

34. 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
Data Structure

35. 2. Binary Tree – Tree Examples
Huffman Code
Data Encoding for communication
AGOODMARKET (11 char. * 7 bits = 77 bits)
 (42bits)
Decoding
 AGOODMARKET
Data Structure

36. 2. Binary Tree – Tree Examples
Huffman Code
Another Example
문 자
빈 도 A 2 B 18 C 9 D 30 E F 36
문 자
빈 도
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
3*9=27 36 F
7*36=252
2*36=72
180 계
104
728
240
488
Data Structure

37. 3. General Tree General Tree
a tree in which each node can have an unlimited out-degree
Data Structure

38. 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
Data Structure

39. 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
Data Structure

40. 3. General Tree Deletion in General Tree Deletion of only leaf node
A node cannot be deleted if it has any children
Other rules for deletion…
Data Structure

41. 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
Data Structure

42. 3. General Tree
General Tree to Binary Tree
Data Structure