TREES AND
BINARY TREES
Become Rich
Force Others
to be Poor
Rob
Banks
Stock
Fraud
The class notes are a compilation and edition from many sources. The
instructor does not claim intellectual property or ownership of the
lecture notes.
NATURE VIEW OF A TREE
leaves
branch
es
root
COMPUTER SCIENTIST’S VIEW
root
leaves
branches
nodes
WHAT IS A TREE




A tree is a finite nonempty
set of elements.
It is an abstract model of a
hierarchical structure.
consists of nodes with a
parent-child relation.
Applications:



Organization charts
File systems
Programming
environments
Computers”R”Us
Sales
US
Europe
Manufacturing
International
Asia
Laptops
Canada
Desktops
R&D
TREE TERMINOLOGY










Root: node without parent (A)
Siblings: nodes share the same parent
Internal node: node with at least one
child (A, B, C, F)
External node (leaf ): node without
children (E, I, J, K, G, H, D)
Ancestors of a node: parent,
grandparent, grand-grandparent, etc.
Descendant of a node: child,
grandchild, grand-grandchild, etc.
Depth of a node: number of ancestors
Height of a tree: maximum depth of
any node (3)
Degree of a node: the number of its
children
Degree of a tree: the maximum
number of its node.
Subtree: tree consisting of a
node and its descendants
A
B
E
F
I
D
C
J
G
K
H
subtree
TREE PROPERTIES
A
C
B
D
E
F
G
H
I
Property
Number of nodes
Height
Root Node
Leaves
Interior nodes
Ancestors of H
Descendants of B
Siblings of E
Right subtree of A
Degree of this tree
Value
TREE ADT


We use positions to abstract
nodes
Generic methods:
integer size()
 boolean isEmpty()
 objectIterator elements()
 positionIterator positions()


Accessor methods:
position root()
 position parent(p)
 positionIterator children(p)

Query methods:
boolean isInternal(p)
boolean isExternal(p)
boolean isRoot(p)
Update methods:
swapElements(p, q)
object replaceElement(p, o)
Additional update methods may
be defined by data structures
implementing the Tree ADT
INTUITIVE REPRESENTATION OF TREE
NODE
List Representation
( A ( B ( E ( K, L ), F ), C ( G ), D ( H ( M ), I, J ) ) )
The root comes first, followed by a list of links to sub-trees
How many link fields are needed in
such a representation?
Data
Link 1
Link 2
…
Link n
TREES

Every tree node:


object – useful information
children – pointers to its children
Data
Data

Data


Data


Data

Data


Data



A TREE REPRESENTATION

A node is represented by
an object storing
Element
 Parent node
 Sequence of children
nodes


B


A
D
F
B
D
A
C
F
E

C

E
LEFT CHILD, RIGHT SIBLING
REPRESENTATION
Data
Left
Child
Right
Sibling
A
B
E
J
F
K
C
D
G
H
L
I
TREE TRAVERSAL

Two main methods:
Preorder
 Postorder


Recursive definition

Preorder:
visit the root
 traverse in preorder the children (subtrees)


Postorder


traverse in postorder the children (subtrees)
visit the root
PREORDER TRAVERSAL



A traversal visits the nodes of a
tree in a systematic manner
Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w)
In a preorder traversal, a node is
visited before its descendants
Application: print a structured
document
1
Become Rich
2
5
1. Motivations
9
2. Methods
3
4
1.1 Enjoy
Life
1.2 Help
Poor Friends
6
2.1 Get a
CS PhD
7
2.2 Start a
Web Site
3. Success Stories
8
2.3 Acquired
by Google
POSTORDER TRAVERSAL


In a postorder traversal, a node is
visited after its descendants
Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v)
Application: compute space used
by files in a directory and its
subdirectories
9
cs16/
3
7
homeworks/
1
h1c.doc
h1c.doc
3K
3K
8
todo.txt
1K
programs/
2
h1nc.doc
h1nc.doc
2K
2K
4
DDR.java
DDR.java
10K
10K
5
Stocks.java
Stocks.java
25K
25K
6
Robot.java
Robot.java
20K
20K
BINARY TREE

A binary tree is a tree with the
following properties:
Applications:
arithmetic expressions
decision processes
searching
Each internal node has at most two
children (degree of two)
 The children of a node are an ordered
pair


We call the children of an internal
node left child and right child

Alternative recursive definition: a
binary tree is either


a tree consisting of a single node, OR
a tree whose root has an ordered pair
of children, each of which is a binary
tree
A
B
C
D
E
H
F
I
G
BINARYTREE ADT


The BinaryTree ADT
extends the Tree ADT,
i.e., it inherits all the
methods of the Tree ADT
Additional methods:
position leftChild(p)
 position rightChild(p)
 position sibling(p)


Update methods may be
defined by data structures
implementing the
BinaryTree ADT
Examples of the Binary Tree
Complete Binary Tree
Skewed Binary Tree
A
B
1
A
A
2
B
B
C
C
3
D
E
D
E
5
4
H
I
F
G
DIFFERENCES BETWEEN A TREE AND 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.
DATA STRUCTURE FOR BINARY TREES

A node is represented
by an object storing

Element
 Parent node
 Left child node
 Right child node

B


A
B
A

D
C
D
E

C


E
ARITHMETIC EXPRESSION TREE

Binary tree associated with an arithmetic expression
internal nodes: operators
 external nodes: operands


Example: arithmetic expression tree for the expression (2  (a
- 1) + (3  b))
+


-
2
a
3
1
b
DECISION TREE

Binary tree associated with a decision process



internal nodes: questions with yes/no answer
external nodes: decisions
Example: dining decision
Want a fast meal?
No
Yes
How about coffee?
On expense account?
Yes
No
Yes
Starbucks
Spike’s
Al Forno
No
Café Paragon
Maximum Number of Nodes in a
Binary Tree
The maximum number of nodes on depth i of a binary tree is 2i, i>=0.
The maximum nubmer of nodes in a binary tree of height k is 2k+1-1,
k>=0.
Prove by induction.
k
i
k +1
2

2
-1

i 0
FULL BINARY TREE

A full binary tree of a given height k has 2k+1–1 nodes.
Height 3 full binary tree.
LABELING NODES IN A FULL BINARY TREE
Label the nodes 1 through 2k+1 – 1.
 Label by levels from top to bottom.
 Within a level, label from left to right.

1
2
3
4
8
6
5
9
10
11
12
7
13
14
15
NODE NUMBER PROPERTIES
1
2
3
4
8
6
5
9
10
11
12
7
13
14
Parent of node i is node i / 2, unless i = 1.
 Node 1 is the root and has no parent.

15
NODE NUMBER PROPERTIES
1
2
3
4
8
6
5
9
10
11
12
7
13
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
7
13
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.

Complete Binary Trees
A labeled binary tree containing the labels 1 to n with root 1, branches
leading to nodes labeled 2 and 3, branches from these leading to 4, 5 and
6, 7, respectively, and so on.
A binary tree with n nodes and level k is complete iff its nodes
correspond to the nodes numbered from 1 to n in the full binary tree of
level k.
1
1
2
4
8
5
9
2
3
6
Complete binary tree
5
4
7
8
3
9
10 11
6
7
12 13 14 15
Full binary tree of depth 3
Binary Tree Traversals
Let l, R, and r stand for moving left, visiting
the node, and moving right.
There are six possible combinations of traversal
lRr, lrR, Rlr, Rrl, rRl, rlR
Adopt convention that we traverse left before
right, only 3 traversals remain
lRr, lrR, Rlr
inorder, postorder, preorder
INORDER TRAVERSAL

In an inorder traversal a node
is visited after its left subtree
and before its right subtree
Algorithm inOrder(v)
if isInternal (v)
inOrder (leftChild (v))
visit(v)
if isInternal (v)
inOrder (rightChild (v))
6
2
8
1
4
3
7
5
9
PRINT ARITHMETIC EXPRESSIONS

Specialization of an inorder
traversal
print operand or operator when
visiting node
print “(“ before traversing left
subtree
print “)“ after traversing right
subtree



+


-
2
a
3
1
b
Algorithm inOrder (v)
if isInternal (v){
print(“(’’)
inOrder (leftChild
(v))}
print(v.element ())
if isInternal (v){
inOrder (rightChild
(v))
print (“)’’)}
((2  (a - 1)) + (3  b))
EVALUATE ARITHMETIC EXPRESSIONS


recursive method returning
the value of a subtree
when visiting an internal
node, combine the values of
the subtrees
Algorithm evalExpr(v)
if isExternal (v)
return v.element ()
else
x  evalExpr(leftChild
(v))
y
evalExpr(rightChild (v))
  operator stored at v
return x  y
+


-
2
5
3
1
2
CREATIVITY:
PATHLENGTH(TREE) =  DEPTH(V) V  TREE
Algorithm pathLength(v, n)
Input: a tree node v and an initial value n
Output: the pathLength of the tree with root v
Usage: pl = pathLength(root, 0);
if isExternal (v)
return n
return
(pathLength(leftChild (v), n + 1) +
pathLength(rightChild (v), n + 1) + n)
EULER TOUR TRAVERSAL

Generic traversal of a binary tree

Includes a special cases the preorder, postorder and inorder traversals

Walk around the tree and visit each node three times:
 on the left (preorder)
 from below (inorder)
 on the right (postorder)
+
L
2


R
B
5
3
1
2
EULER TOUR TRAVERSAL
eulerTour(node v) {
perform action for visiting node on the left;
if v is internal then
eulerTour(v->left);
perform action for visiting node from below;
if v is internal then
eulerTour(v->right);
perform action for visiting node on the right;
}
TREE
BINARY TREE
 sebuah
pengorganisasian secara hirarki
dari beberapa buah simpul, dimana
masing-masing simpul tidak mempunyai
anak lebih dari 2.
 Simpul yang berada di bawah sebuah
simpul dinamakan anak dari simpul
tersebut.
 Simpul yang berada di atas sebuah
simpul dinamakan induk dari simpul
tersebut.
BINARY TREE
ISTILAH DALAM TREE
Term
Node
Parent
Definition
Sebuah elemen dalam sebuah tree; berisi sebuah informasi
Node yang berada di atas node lain secara langsung; B adalah
parent dari D dan E
Child
Cabang langsung dari sebuah node; D dan E merupakan children
dari B
Root
Node teratas yang tidak punya parent
Sibling
Sebuah node lain yang memiliki parent yang sama; Sibling dari
B adalah C karena memiliki parent yang sama yaitu A
Leaf
Sebuah node yang tidak memiliki children. D, E, F, G, I adalah
leaf. Leaf biasa disebut sebagai external node, sedangkan node
selainnya disebut sebagai internal node. B, A, C, H adalah
internal node
Level
Semua node yang memiliki jarak yang sama dari root.
Alevel 0; B,Clevel 1; D,E,F,G,Hlevel 2; Ilevel 3
Depth
Jumlah level yang ada dalam tree
Complete Semua parent memiliki children yang penuh
Balanced Semua subtree memiliki depth yang sama
STRUKTUR BINARY TREE
 Masing-masing
simpul dalam binary tree
terdiri dari tiga bagian yaitu sebuah data
dan dua buah pointer yang dinamakan
pointer kiri dan kanan.
DEKLARASI TREE
typedef char typeInfo;
typedef struct Node tree;
struct Node {
typeInfo info;
tree *kiri;
/* cabang kiri */
tree *kanan; /* cabang kanan */
};
PEMBENTUKAN TREE
 Dapat
dilakukan dengan dua cara :
rekursif dan non rekursif
 Perlu memperhatikan kapan suatu node
akan dipasang sebagai node kiri dan
kapan sebagai node kanan.
 Misalnya ditentukan, node yang berisi
info yang nilainya “lebih besar” dari
parent akan ditempatkan di sebelah
kanan dan yang “lebih kecil” di sebelah
kiri.
 Sebagai contoh jika kita memiliki
informasi “HKACBLJ” maka pohon biner
yang terbentuk
PEMBENTUKAN TREE
PEMBENTUKAN TREE
Langkah-langkah Pembentukan Binary Tree
1. Siapkan node baru
- alokasikan memory-nya
- masukkan info-nya
- set pointer kiri & kanan = NULL
2. Sisipkan pada posisi yang tepat
- penelusuran  utk menentukan posisi yang
tepat; info yang nilainya lebih besar dari parent
akan ditelusuri di sebelah kanan, yang lebih kecil
dari parent akan ditelusuri di sebelah kiri
- penempatan  info yang nilainya lebih dari
parent akan ditempatkan di sebelah kanan, yang
lebih kecil di sebelah kiri
METODE TRAVERSAL
 Salah
satu operasi yang paling umum dilakukan
terhadap sebuah tree adalah kunjungan (traversing)
 Sebuah kunjungan berawal dari root, mengunjungi
setiap node dalam tree tersebut tepat hanya sekali

Mengunjungi artinya memproses data/info yang ada pada
node ybs
 Kunjungan
bisa dilakukan dengan 3 cara:
1. Preorder
2. Inorder
3. Postorder
 Ketiga macam kunjungan tersebut bisa dilakukan
secara rekursif dan non rekursif
PREORDER

Kunjungan preorder, juga disebut dengan depth
first order, menggunakan urutan:
Cetak isi simpul yang dikunjungi
Kunjungi cabang kiri
Kunjungi cabang kanan
PREORDER
void preorder(pohon ph)
{
if (ph != NULL)
{
printf("%c ", ph->info);
preorder(ph->kiri);
preorder(ph->kanan);
}
}
PREORDER
A
B
D
C
E
F
ABDECFG
G
INORDER
Kunjungan secara inorder, juga sering disebut
dengan symmetric order, menggunakan urutan:
 Kunjungi cabang kiri
 Cetak isi simpul yang dikunjungi
 Kunjungi cabang kanan

INORDER
void inorder(pohon ph)
{
if (ph != NULL)
{
inorder(ph->kiri);
printf("%c ", ph->info);
inorder(ph->kanan);
}
}
INORDER
A
B
D
C
E
F
DBEAFCG
G
POSTORDER
Kunjungan secara postorder menggunakan
urutan:
 Kunjungi cabang kiri
 Kunjungi cabang kanan
 Cetak isi simpul yang dikunjungi

POSTORDER
void postorder(pohon ph)
{
if (ph != NULL)
{
postorder(ph->kiri);
postorder(ph->kanan);
printf("%c ", ph->info);
}
}
POSTORDER
A
B
D
C
E
F
DEBFGCA
G
REFERENCES

http://www.cse.unt.edu/~huangyan/3110/Lectures
/Trees.ppt