Review of Chapter 5

張啟中

Definition of Tree A tree is a finite set of one or more nodes A tree is a finite set of one or more nodes

such that such that A specially designated node called root. The remaining nodes are partitioned into n0

disjoint sets T1, …, Tn, called subtrees.

root

T1 T2 Tn

Terminology

A

B C D

E F G H I J

L M

node

leaf

parentB

childrenE

siblingsB, D

1

2

3

4

levelroot

K ancestorsA, D, H

height or depth = maximum level of any node in the tree = 4

degree3

degree of a tree = maximum of the degree of the nodes = 3

Representation of Trees

Tree 的表示方法，常用的有下列三種 List Representation Left Child-Right Sibling Representation Representation as a Degree-Two Tree

Representation of Trees (1) List Representation

利用 generalized lists 來表示 (A( B(E(K, L), F), C(G), D(H(M), I, J ) ) ) 利用 fixed size 的 Node

A

B F 0

E K L 0

C G 0

0

本圖未完成，留給同學練習

Representation of Trees (1)

Lemma 5.1If T is a k-ary tree (i.e., a tree of degree k) with n nodes, each having a fixed size as in Figure 5.4, then n(k-1) + 1 of the nk child fileds are 0, n ≥ 1.

Data Child 1 Child 2 Child 3 Child 4 … Child k

Wasting memory!

Representation of Trees (2)

A

B C D

E F G H I J

L MK

data

left child Right sibling

Representation of Trees (3)A

B

C

D

E

F G

H

I

J

L

M

K

Relation of Tree Data Structures

Binary Tree

Complete Binary Tree Binary Search Tree

Search Struct

Max Heap Tree Winner

Max PQ

Binary Trees A binary tree is a finite set of nodes that is ei

ther empty or consists of a root and two disjoint binary trees called the left subtree and the right subtree.

A

B C

D

EE

Binary Trees VS. Trees

category

itemTrees Binary Trees

The order of the subtrees nonedistinctions between

left and right

Empty (zero nodes) No Yes

degree 0..n 0..2 ( 註 )

註：修正老師上課時的說法

Full Binary Tree A full binary tree of depth k is a binary tree

of depth k having 2k – 1 nodes, k ≥ 0.

A

B C

D

IH

G

M

1

2

3

4

level

E F

J K L N O

Complete Binary Tree A binary tree with n nodes and depth k is complete if and only if its nodes correspond to the nodes numbered from 1 to n in the full binary tree of depth k.

1

2

3

4

level

A

B C

D GFE

IH

Properties of Binary Trees

Lemma 5.2 [Maximum number of nodes] The maximum number of nodes on level i of a

binary tree is 2i-1, i ≥ 1. The maximum number of nodes in a binary tree of

depth k is 2k – 1, k ≥ 1. Lemma 5.3 [Relation between number of leaf nodes and

nodes of degree 2] For any non-empty binary tree, T, if n0 is the

number of leaf nodes and n2 the number of nodes of degree 2, then n0 = n2 + 1.

Properties of Binary Trees

Lemma 5.4

If a complete binary tree with n nodes is represented sequentially, then for any node with index i, 1 ≤ i ≤ n, we have: parent(i) is at if i ≠1. If i = 1, i is at the root and has no

parent. left_child(i) is at 2i if 2i ≤ n. If 2i > n, then i has no left child. right_child(i) is at 2i + 1 if 2i + 1 ≤ n. If 2i + 1 > n, then i has n

o right child.

Position zero of the array is not used.

2

i

Binary Tree Representations

Array Representation Use Lemma 5.4

Linked Representation

Binary Tree Representations (Array)

A

B

C

D

E

—

A

B

—

C

—

—

—

D

—

E

0

1

2

3

4

5

6

7

8

9

16

skewed binary tree

Binary Tree Representations (Array)

A

B C

D GFE

IH

—

A

B

C

D

E

F

G—

—

—

0

1

2

3

4

5

6

7

8

9

16

Binary Tree Representations (Linked)

A

B

C

D

E

LeftChild data RightChilddata

LeftChild Right ChildA 0

B 0

C 0

D 0

0 E 0

root

Binary Tree Representations (Linked)

A

B C

D E

H 0 I 0

F 0 G 0

root

Binary Tree Representations (Linked)class Tree; //forward declaration

class TreeNode {

friend class Tree;

private:

TreeNode *LeftChild;

char data;

TreeNode *RightChild;

};

class Tree {

public:

//Tree operation

private:

TreeNode *root;

};

Manipulation of Binary Tree Traversal

Inorder (LVR) (Stack) Postorder (LRV) (Stack) Preorder (VLR) (Stack) Level-Order (Queue)

Copying Binary Trees Testing Equality

Two binary trees are equal if their topologies are the same and the information in corresponding nodes is identical.

Binary Tree Traversal ( Inorder LVR )

+

* E

* D

/ C

A B A / B * C * D + E

Binary Tree Traversal ( Postorder LRV )

+

* E

* D

/ C

A B A B / C * D * E +

Binary Tree Traversal ( Preorder VLR )

+

* E

* D

/ C

A B

+ * * / A B C D E

Binary Tree Traversal ( Level-Order )

+

* E

* D

/ C

A B

+ * E * D / C A B

Implement of Binary Tree Traversal Recursive Method

Program 5.1 (p263) --- Inorder Program 5.2 (p264) --- Preorder Program 5.3 (p265) --- Postorder

Iterative Method Program 5.4 (p266) --- Inorder Program 5.5 and 5.6 --- Use Iterator (Inorder)

Implement of Binary Tree Traversal Recusivevoid Tree :: inorder() // Driver calls workhorse for traversal of entire tree. The // driver is declared as a public member function of Tree. { inorder(root); }

void Tree :: inorder(TreeNode *CurrentNode)// Workhorse traverses the subtree rooted at CurrentNode // The workhorse is declared as a private member function of Tree. { if(CurrentNode){ inorder(CurrentNode->LeftChild); cout << CurrentNode->data; inorder(CurrentNode->RightChild); } }

Implement of Binary Tree Traversal Iterativevoid Tree::NonrecInorder()//nonrecursive inorder traversal using a stack{

Stack<TreeNode *> s; //declare and initialize stackTreeNode *CurrentNode = root;while (1) {

while (CurrentNode) { //move down LeftChild fields s.Add(CurrentNode); //add to stack CurrentNode = CurrentNode->LeftChild;

}if (!s.IsEmpty()) { //stack is not empty

CurrentNode = *s.Delete(CurrentNode); cout << CurrentNode->data << endl; CurrentNode = CurrentNode->RightChild;

}else break;

}}

Implement of Binary Tree Traversalvoid Tree::LevelOrder()//Traverse the binary tree in level order{Queue<TreeNode *> q;TreeNode *CurrentNode = root;while (CurrentNode) {

cout << CurrentNode->data<<endl;if (CurrentNode->LeftChild)

q.Add(CurrentNode->LeftChild);if (CurrentNode->RightChild)

q.Add(CurrentNode->RightChild);CurrentNode = *q.Delete();

}}

Traversal without Stack

二種方式 每個 node 都增加一個欄位紀錄 parent node 的

位置。 將 binary trees 改成 threaded binary Trees.