Download Binary Tree - Universitas Muhammadiyah Malang

Algoritma dan Struktur Data
Binary Tree
Teknik Informatika
Universitas Muhammadiyah Malang
SP Ganjil - 2013
Tujuan Instruksional
• Mahasiswa mampu menjelaskan mengenai
konsep Tree
• Mahasiswa mampu membuat dan
mendeklarasikan struktur data Tree
• Mahasiswa mampu menerapkan dan
mengimplementasikan Tree
Sub Topik
•
•
•
•
•
Penjelasan Tree
Istilah pada tree
Binary Tree
Jenis Binary Tree
ADT Binary tree
Linear Lists And Trees
• Linear lists are useful for serially ordered data.
– (e0, e1, e2, …, en-1)
– Days of week.
– Months in a year.
– Students in this class.
• Trees are useful for hierarchically ordered data.
– Employees of a corporation.
• President, vice presidents, managers, and so on.
– Java’s classes.
• Object is at the top of the hierarchy.
• Subclasses of Object are next, and so on.
Hierarchical Data And Trees
• The element at the top of the hierarchy is the
root.
• Elements next in the hierarchy are the children
of the root.
• Elements next in the hierarchy are the
grandchildren of the root, and so on.
• Elements that have no children are leaves.
Istilah pada Tree
Java’s Classes (Part Of Figure 1.1)
Object
root
children of root
Number
Integer
Double
Throwable
Exception
OutputStream
FileOutputStream
grand children of
root
RuntimeException
great grand child of root
Definition
• A tree t is a finite nonempty set of elements.
• One of these elements is called the root.
• The remaining elements, if any, are
partitioned into trees, which are called the
subtrees of t.
Subtrees
Object
Number
Integer
Double
root
Throwable
OutputStream
Exception
FileOutputStream
RuntimeException
Leaves
Object
Number
Integer
Double
Throwable
OutputStream
Exception
FileOutputStream
RuntimeException
Parent, Grandparent, Siblings, Ancestors, Descendants
Object
Number
Integer
Double
Throwable
OutputStream
Exception
FileOutputStream
RuntimeException
Levels
Object
Number
Integer
Double
Level
1
Level 2
Throwable
OutputStream
Exception
FileOutputStream
Level 3
RuntimeException
Level 4
Caution
• Some texts start level numbers at 0 rather than
at 1.
• Root is at level 0.
• Its children are at level 1.
• The grand children of the root are at level 2.
• And so on.
• We shall number levels with the root at level 1.
height = depth = number of levels
Object
Number
Integer
Double
Level
1
Level 2
Throwable
OutputStream
Exception
FileOutputStream
Level 3
RuntimeException
Level 4
Node Degree = Number Of Children
Object
2
1
Number
0
Integer
0
Double
3
Throwable
1
1
Exception
RuntimeException
OutputStream
0
FileOutputStream
0
Tree Degree = Max Node Degree
Object
2
1
Number
0
Integer
0
Double
3
Throwable
1
Exception
RuntimeException
Degree of tree = 3.
1
OutputStream
0
FileOutputStream
0
Latihan
Ancestor (F)?
Predesesor (F) ?
Descendant (B)?
Succesor (B)?
Parent (I)?
Child (C)?
Sibling (G)?
Size?
Height?
Root?
Leaf?
Degree (C)?
Latihan
• Jawaban :
Ancestor (F) = C,A
Predesesor (F) = A,B,C
Descendant (B) = D atau E
Sucessor (B) = D,E,F,G,H
Parent (I) = H
Child (C) = F,G,H
Sibling (G) = F,H
Size = 9
Height = 4
Root = A
Leaf = D,E,F,G,I
Degree (C) = 3
Latihan
Gambarkan tree dari representasi berikut :
DF
DB
KJ
KL
BA
BC
H D
HK
FE
FG
JI
Tentukan :
1. Root
2. Leaf
3. Heigth
4. Child H
5. Parent A
Binary Tree
Binary Tree
• Finite (possibly empty) collection of elements.
• A nonempty binary tree has a root element.
• The remaining elements (if any) are partitioned
into two binary trees.
• These are called the left and right subtrees of
the binary tree.
Differences Between A Tree & A Binary Tree
• No node in a binary tree may have a degree
more than 2, whereas there is no limit on
the degree of a node in a tree.
• A binary tree may be empty; a tree cannot
be empty.
Differences Between A Tree & A Binary Tree
• The subtrees of a binary tree are ordered;
those of a tree are not ordered.
a
b
a
b
• Are different when viewed as binary trees.
• Are the same when viewed as trees.
Contoh Binary Tree
• Representasi ekspresi arithmatik
Minimum Number Of Nodes
• Minimum number of nodes in a binary tree
whose height is h.
• At least one node at each of first h levels.
minimum number of nodes is h
Maximum Number Of Nodes
• All possible nodes at first h levels are present.
Maximum number of nodes
= 1 + 2 + 4 + 8 + … + 2h-1
= 2h - 1
Number Of Nodes & Height
• Let n be the number of nodes in a binary tree
whose height is h.
• h <= n <= 2h – 1
• log2(n+1) <= h <= n
Full Binary Tree
• A full binary tree of a given height h has 2h – 1
nodes.
Height 4 full binary tree.
Numbering Nodes In A Full Binary
Tree
• Number the nodes 1 through 2h – 1.
• Number by levels from top to bottom.
• Within a level number from left to right.
1
2
3
4
8
6
5
9
10
11
12
13
7
14
15
Node Number Properties
1
2
3
4
8
6
5
9
10
11
12
13
7
14
15
• Parent of node i is node i / 2, unless i = 1.
• Node 1 is the root and has no parent.
Node Number Properties
1
2
3
4
8
6
5
9
10
11
12
13
7
14
15
• Left child of node i is node 2i, unless 2i > n,
where n is the number of nodes.
• If 2i > n, node i has no left child.
Node Number Properties
1
2
3
4
8
6
5
9
10
11
12
13
7
14
15
• Right child of node i is node 2i+1, unless 2i+1 >
n, where n is the number of nodes.
• If 2i+1 > n, node i has no right child.
Akses Elemen untuk Start Level 0
• Asumsi root dimulai dari index 0 :
– Anak kiri dari node i berada pada indeks : 2*i+1
– Anak kanan dari node i berada pada indeks : 2*i+2
Struktur Data - Tree
33
Representasi Binary Tree
• Binary tree dapat direpresentasikan dengan
menggunakan array maupun linked list.
Representasi Tree
Representasi tree menggunakan array (asumsi root pada index 0) :
H
D
K
B
F
J
L
A
C
E
G
I
0
1
2
3
4
5
6
7
8
9
10
11
Linked Representation
root
H
D
K
J
B
L
F
I
A
C
E
leftChild
element
G
rightChild
The Class BinaryTreeNode
package dataStructures;
public class BinaryTreeNode
{
Object element;
BinaryTreeNode leftChild; // left subtree
BinaryTreeNode rightChild;// right subtree
// constructors and any other methods
// come here
}
Array Representation
a
1
2
b
4
3
c
5
e
d
8
h
tree[]
i
9
6
f
10
j
a b c d e f g h i j
0
5
10
7
g
Latihan
a1
b 3
7
c
d
15
Right-Skewed Binary Tree
a1
b 3
7
c
d
a -
tree[]
0
15
b - - - c - - - - - - - d
5
10
15
Some Binary Tree Operations
•
•
•
•
•
Determine the height.
Determine the number of nodes.
Make a clone.
Determine if two binary trees are clones.
Display the binary tree.
Binary Tree Traversal
Definisi
• Penelusuran seluruh node pada binary tree.
• Metode :
– Preorder
– Inorder
– Postorder
– Level order
PreOrder Traversal
•
Preorder traversal
1. Cetak data pada root
2. Secara rekursif mencetak seluruh data pada
subpohon kiri
3. Secara rekursif mencetak seluruh data pada
subpohon kanan
Preorder Example (visit = print)
a
b
c
a b c
Preorder Example (visit = print)
a
b
f
e
d
g
c
h
i
a b d g h e i c f j
j
Preorder Of Expression Tree
/
+
*
e
+
a
b
c
d
/ * + a b - c d + e f
Gives prefix form of expression!
f
InOrder Traversal
• Inorder traversal
1.Secara rekursif mencetak seluruh data pada
subpohon kiri
2.Cetak data pada root
3.Secara rekursif mencetak seluruh data pada
subpohon kanan
Inorder Example (visit = print)
a
b
c
b a c
Inorder Example (visit = print)
a
b
f
e
d
g
c
h
i
g d h b e i a f j c
j
Postorder Traversal
• Postorder traversal
1.Secara rekursif mencetak seluruh data pada
subpohon kiri
2.Secara rekursif mencetak seluruh data pada
subpohon kanan
3.Cetak data pada root
Postorder Example (visit = print)
a
b
c
b c a
Postorder Example (visit = print)
a
b
f
e
d
g
c
h
i
g h d i e b j f c a
j
Postorder Of Expression Tree
/
+
*
e
+
a
b
c
d
a b + c d - * e f + /
Gives postfix form of expression!
f
Traversal Applications
a
b
f
e
d
g
c
h
• Make a clone.
• Determine height.
•Determine number of nodes.
i
j
Latihan
• Telusuri pohon biner berikut dengan
menggunakan metode pre, in, post, dan level
traversal.
Latihan 1
+
a.
3
*
b.
5
-
2
/
8
4
Latihan 2
Level-Order Example (visit = print)
a
b
f
e
d
g
c
h
i
a b c d e f g h i j
j
Contoh : Pohon Ekspresi
Implementasi Binary Tree
Operasi pada Binary Tree
• Operas-operasi yang ada pada binary tree
adalah :
1. Deklarasi
2. Pengecekkan kosong (isEmpty)
3. Penambahan node
4. Traversal (penelusuran node)
Deklarasi dengan Array
• Dengan menggunakan array, prosesnya meliputi :
1. Deklarasi class ArrayBinaryTree
2. Deklarasi variabel array
3. Instansiasi variabel array
variabel array digunakan untuk menyimpan node-node yang
membentuk binary tree. Yaitu berupa variabel array 1
dimensi. Tipe variabel array akan menentukan jenis data
yang dapat disimpan pada binary tree.
4. Deklarasi variabel size (untuk menyimpan jumlah
node)
5. Deklarasi variabel last (untuk menyimpan index
node terakhir)
Contoh program
• Deklarasi :
public class ArrayBinaryTree //deklarasi class
{
static Object [] a; //deklarasi array
static int last; //deklarasi last
static int size=0; //deklarasi size
public static void main(String [] args)
{
a = new Integer [15]; //instansiasi*
.....
.....
*contoh : instansiasi variabel array dengan panjang 15 element.
isEmpty dengan Array
• Digunakan untuk mengecek binary tree dalam
kondisi kosong atau terisi node.
• Pengecekan menggunakan variabel size.
• Mengembalikan true jika size =0.
• Mengembalikan false jika nilai size > 0.
Contoh program
• isEmpty :
static boolean isEmpty()
{
return (size==0);
}
Penambahan Node dengan Array
• Penambahan node dengan menggunakan
index.
• Ketika terjadi penambahan node terjadi
increment nilai pada variabel size. Dan
variabel last akan menunjuk index node yang
terakhir kali ditambahkan.
Contoh program
• Penambahan node :
static void addNode(int i, int isi)
{
a[i] = new Integer(isi);
size++;
last = i;
}
Traversal dengan Array
• Penelusuran (traversal) digunakan untuk
menelusuri node pada binary tree satu persatu.
• Terdiri dari 3 metode traversal :
– Pre-order
– In-order
– Post-order
Contoh program
• Penelusuran node (inorder):
public static void inOrder(Object [] theArray, int theLast)
{
a = theArray;
if(!isEmpty())
{
theInOrder(1);
}
else
System.out.println("Binary Tree Kosong");
}
Contoh program
• Program rekursif untuk penelusuran subtree :
static void theInOrder(int i)
{
if (i <= last && a[i]!=null)
{
theInOrder(2 * i);
visit(i);
theInOrder(2 * i + 1);
}
}
Kunjungan Node
• Digunakan untuk mengunjungi node. Proses
yang dilakukan adalah mencetak node untuk
menandai bahwa node tersebut sudah
dikunjungi.
Contoh program
• Kunjungan/visit node :
public static void visit(int i)
{
System.out.print(a[i] + " ");
}
Deklarasi dengan Linked list
• Dengan menggunakan linked list, prosesnya
meliputi :
1. Pembuatan class node (double linked list)
2. Pembuatan class LinkedBinaryTree
3. Deklarasi variabel root bertipe Node
4. Deklarasi variabel size
Contoh program
• Deklarasi :
public class LinkedBTree{
static Node2P root;
static int size=0;
.....
}
isEmpty
• Digunakan untuk mengecek binary tree dalam
kondisi kosong atau tidak.
• Pengecekan dilakukan pada variabel root.
• Jika root menunjuk null, kondisi kosong dan
mengembalikan nilai true. Sebaliknya, jika
root tidak menunjuk null berarti kondisi binary
tree tidak kosong dan mengembalikan nilai
false.
Contoh program
• Pengecekan kosong:
static boolean isEmpty()
{
return (root==null);
}
Penambahan Node
• Untuk melakukan penambahan node, terlebih
dahulu harus di-create node baru.
Node x = new Node(data);
• Ketika ada penambahan node terjadi
increment nilai pada variabel size.
Contoh program
• Penambahan node :
static void addNode(Node2P baru, Node2P kiri, Node2P kanan)
{
baru.next = kanan;
baru.previous = kiri;
size++;
}
setRoot
• Digunakan untuk menandai node mana yang
dijadikan sebagai root.
static void setRoot(Node2P r)
{
root = r;
}
Traversal dengan linkedlist
• Penelusuran (traversal) digunakan untuk
menelusuri node pada binary tree satu persatu.
• Terdiri dari 3 metode traversal :
– Pre-order
– In-order
– Post-order
Contoh program
• Penelusuran node (postOrder) :
static void postOrder()
{
if(!isEmpty())
thePostOrder(root);
else
System.out.println("Binary Tree Kosong");
}
Contoh program
• Program rekursif untuk penelusuran subtree :
static void thePostOrder(Node2P node)
{
if(node != null)
{
thePostOrder(node.previous);
thePostOrder(node.next);
System.out.print(node.data + "
}
}
");
Contoh program
• Penelusuran node (level order) :
static void theLevelOrder(Node2P node)
{
QueueArray temp = new QueueArray();
temp.inisialisasi(15);
temp.enqueue(node);
while(temp.jumlah_item >0)
{
if(node.previous !=null)
temp.enqueue(node.previous);
if(node.next !=null)
temp.enqueue(node.next);
System.out.print(node.data + " ");
if(!temp.isEmpty())
{
temp.dequeue();
node = temp.peekQueue();
}
}
}
Implementasi Binary Search Tree
Binary Search Tree (BST)
• Binary Search Tree (pohon telusur biner)
• Disebut juga Ordered Binary tree yaitu binary
tree yang seluruh children dari tiap node
terurut.
• Data pada subtree kiri lebih kecil dari data
pada subtree kanan.
Contoh Binary Search Tree
• Root = 20
20
10
6
2
40
15
8
30
25
• Bagaimana jika ada penambahan node 13?
• Bagaimana jika ada penambahan node 50?
• Bagaimana jika ada penambahan node 26?
Latihan
• Buatlah sebuah Binary Search Tree
berdasarkan proses-proses berikut :
insert 17, insert 8, insert 32, insert 6, insert
10, insert 21, insert 45, insert 1, insert 7,
insert 9, insert 26, insert 37, insert 50.
Operasi Binary Search Tree
• Operasi-operasi yang dilakukan pada binary
search tree meliputi :
1. Penambahan node
2. Penghapusan node
3. Pencarian node
Penambahan BST
• Penambahan node pada BST harus mengikuti
aturan minMax, dimana node yang bernilai
lebih kecil dari root diletakkan pada subtree
sebelah kiri sedangkan node yang bernilai
lebih besar diletakkan pada subtree sebelah
kanan. Jika ada nilai yang sama maka node
tersebut di-overwrite.
Contoh program
• Penambahan :
Node2P insert(int x, Node2P t)
{
if (t == null) {
t = new Node2P (x, null, null);
} else if (x < t.data) {
t.previous = insert (x, t.previous);
} else if (x > t.data) {
t.next = insert (x, t.next);
} else {
t=t;
}
return t;
}
Penghapusan BST
Ada 3 kasus :
 Elemen ada di leaf/daun.
 Elemen yang memiliki degree 1.
 Elemen yang memiliki degree 2.
Penghapusan Node Daun (Node 7)
20
10
6
2
40
15
8
30
18
25
35
7
Remove a leaf element. key = 7
Penhapusan Node Daun (Node 35)
20
10
6
2
40
15
8
30
18
25
35
7
Remove a leaf element. key = 35
Penghapusan Node Ber-degree 1
20
10
6
2
40
15
8
30
18
25
35
7
Remove from a degree 1 node. key = 40
Penghapusan Node Ber-degree 1
20
10
6
2
40
15
8
30
18
25
35
7
Remove from a degree 1 node. key = 15
Penghapusan Node Ber-degree 2
20
10
6
2
40
15
8
30
18
25
35
7
Remove from a degree 2 node. key = 10
Remove From A Degree 2 Node
20
10
6
2
40
15
8
30
18
25
35
7
Replace with largest key in left subtree (or
smallest in right subtree).
Penghapusan Node Ber-degree 2
20
10
6
2
40
15
8
30
18
25
35
7
Replace with largest key in left subtree (or
smallest in right subtree).
Penghapusan Node Ber-degree 2
20
8
6
2
40
15
8
30
18
25
35
7
Replace with largest key in left subtree (or
smallest in right subtree).
Latihan
20
10
6
2
40
15
8
30
18
25
35
7
Remove from a degree 2 node. key = 20
Penghapusan Node Ber-degree 2
20
10
6
2
40
15
8
30
18
25
35
7
Replace with largest in left subtree.
Penghapusan Node Ber-degree 2
20
10
6
2
40
15
8
30
18
25
35
7
Replace with largest in left subtree.
Penghapusan Node Ber-degree 2
18
10
6
2
40
15
8
30
18
25
35
7
Replace with largest in left subtree.
Hasil Akhir
18
10
6
2
15
8
7
40
30
25
35
Contoh program
• Penghapusan:
Node2P remove(int x, Node2P t)
{
if (t == null) t=null;
if (x < t.data) {
t.previous = remove(x, t.previous);
} else if (x > t.data) {
t.next = remove(x, t.next);
} else if (t.previous != null && t.next != null) {
t.data = findMin(t.next).data;
t.next = removeMin(t.next);
} else {
t = (t.previous != null) ? t.previous : t.next;
}
return t;
}
Contoh program
• Penghapusan node terkecil :
Node2P removeMin(Node2P t)
{
if (t == null) t=null;
if (t.previous != null) {
t.previous = removeMin (t.previous);
} else {
t = t.next;
}
return t;
}
Contoh program
Pencarian Node terkecil :
Node2P findMin (Node2P t)
{
if (t == null) t=null;
while (t.previous != null) {
t = t.previous;
}
return t;
}
Pustaka
• Sartaj Sahni , “Data Structures & Algorithms”,
Presentation L20-24.
• Mitchell Waite, “Data Structures & Algorithms in
Java”, SAMS, 2001