Chapter 6. Introduction to Trees

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

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 • 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

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

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 nodeis adjacent to the next one 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. • 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

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

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

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

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

2. Binary Tree • Definitions • 임의의노드가 둘 이상의 자식노드를 가지는 트리를일반트리 (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

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

2. Binary Tree • Examples of Binary Trees Data Structure

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): 트리의높이: • 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

2. Binary Tree • Definitions • 삼진트리 (TernaryTree) 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 • : the height of the left subtree • : the heighr of the right subtree • Balance Factor • Balanced Binary Tree • A tree with |B|  1 HL HR Data Structure

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 tree의Height가H일 때 node의 갯수가 Data Structure

2. Binary Tree • Nearly Complete Binary Tree • 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 N일 때 height는 Data Structure

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

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

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

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

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

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

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

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

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

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(")"); } } 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+ 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 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% Data Structure

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 lowestfrequencies • Merge them and make a binarysybtree • Mark the merged frequency intothe root of the subtree • End of repeat Data Structure

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

2. Binary Tree – Tree Examples • Huffman Code • Another Example Data Structure

3. General Tree • General Tree • 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 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 Data Structure

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

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

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

Chapter 6. Introduction to Trees