Download Trees

Department of Computer Eng. & IT
Amirkabir University of Technology
(Tehran Polytechnic)
Data Structures
Trees
Lecturer: Abbas Sarraf ([email protected])
Introduction
level
A
B
E
K
1
C
F
L
G
D
H
M
Figure 5.2 : A sample tree
Dept. of Computer Eng. & IT, Amirkabir University of Technology
I
2
J
3
4
List Representation
A
0
B
E
F 0
C
K
L 0
G0
I
D
H
M0
Figure 5.3 : List representation of the tree of Figure 5.2
Dept. of Computer Eng. & IT, Amirkabir University of Technology
J 0
Left child-right sibling representation
Left child-right sibling representation
node structure
data
left child
right sibling
the left child field of each node points to its leftmost child (if
any), and the right sibling field points to its closest right sibling
(if any)
A
B
E
K
L
Dept. of Computer Eng. & IT, Amirkabir University of Technology
F
D
C
G
H
M
I
J
Representation as a degree-two tree
rotate the right-sibling pointers clockwise by 45 degrees
the two children of a node are referred to as the left and right
children
A
A
B
B
B
C
E
K
F
D
G
L
H
M
I
J
Figure 5.7 : Left child-right child tree representation
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Binary Trees
level
A
B
B
C
D
D
1
A
H
2
C
E
F
G
3
4
I
E
5
(a)
(b)
Figure 5.10 : Skewed and complete binary trees
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Properties of Binary Trees
Lemma 5.2 [Maximum number of nodes]
(1) The max number of nodes on level i of a binary tree is 2i-1, i≥ 1
(2) The max number of nodes in a binary tree of depth k is 2k-1, k≤ 1
A full binary tree of depth k
a binary tree of depth k having 2k-1 nodes, k≥ 0
Dept. of Computer Eng. & IT, Amirkabir University of Technology
1
2
4
8
3
5
9 10 11
6
12
7
13
14
15
Figure 5.11 : Full binary tree of depth 4 with sequential node numbers
Dept. of Computer Eng. & IT, Amirkabir University of Technology
A binary tree with n nodes and depth k is complete
its nodes correspond to the nodes numbered from 1 to n in the full
binary tree of depth k
The height of a complete binary tree with n nodes is log 2 (n  1)
Dept. of Computer Eng. & IT, Amirkabir University of Technology
data
LeftChild
data
RightChild
LeftChild
Figure 5.13 : Node representations
Dept. of Computer Eng. & IT, Amirkabir University of Technology
RightChild
root
A
A
B
D
H
B
C
E
F
G
I
D
0 H 0
C
E 0
0 F 0
0 G 0
0 I 0
(b)
Figure 5.14 : Linked representation for the binary trees of Figure 5.10
Dept. of Computer Eng. & IT, Amirkabir University of Technology
5.3.1 Introduction
Tree traversal
visiting each node in the tree exactly once
a full traversal produces a linear order for the nodes
Order of node visit
L : move left
V : visit node
R : move right
possible combinations : LVR, LRV, VLR,
VRL, RVL, RLV
traverse left before right
LVR : inorder
LRV : postorder
VLR : preorder
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Constructing the original tree from inorder/preorder
Constructing the original tree from inorder/postorder
Constructing the original tree from preorder/postorder
Dept. of Computer Eng. & IT, Amirkabir University of Technology
5.6 Heaps(Cont’)
14
9
12
10
7
8
6
30
3
25
5
6
Figure 5.24 : Max heaps
2
7
10
4
8
11
10
20
6
83
50
Figure 5.25 : Min heaps
Dept. of Computer Eng. & IT, Amirkabir University of Technology
21
Heaps
5.6.3 Insertion into Max Heap
Examples
20
15
2
10
14
(a)
(b)
5
20
21
15
14
15
10
2
(c)
14
20
10
2
(d)
Figure 5.26 : Insertion into a max heap
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Heaps(Cont’)
Implementation
need to move from child to parent
heap is complete binary tree
use formula-based representation
Lemma 5.4 : parent(i) is at i 2if
 i≠ 1
Complexity is O(log n)
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Heaps(Cont’)
template <class Type>
void MaxHeap<Type>::Insert(const Element<Type> &x)
// insert x into the max heap
{
if (n = = MaxSize) {HeapFull(); return;}
n++;
for (int i = n; 1; ) {
if (i = = 1) break; // at root
if (x.key <= heap[i/2].key) break;
// move from parent to i
heap[i] = heap[i/2];
i /= 2;
}
heap[i] = x;
}
Program 5.18 : Insertion into a max heap
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Heaps(Cont’)
Deletion from Max Heap
Example
2
15
15
14
20
10
14
2
10
6
(a)
(b)
Figure 5.27 : Deletion from a heap
Dept. of Computer Eng. & IT, Amirkabir University of Technology
(c)
Heaps(Cont’)
template <class Type>
Element<Type> *MaxHeap<Type>::DeleteMax(Element<Type> &x)
// Delete from the max heap
{
if (!n) {HeapEmpty(); return 0;}
x = heap[1]; Element<Type> k = heap[n]; n--;
for (int i = 1, j = 2; j <= n; )
{
if (j<n) if (heap[j].key < heap[j+1].key) j++;
// j points to the larger child
if (k.key >= heap[j].key) break;
heap[i] = heap[j]; // move child up
i = j; j *= 2;
// move i and j down
}
heap[i] = k;
return &x;
}
Program 5.19 : Deletion from a max heap
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Binary Search Trees
Definition
Definition
binary tree which may be empty
if not empty
(1) every element has a distinct key
(2) keys in left subtree < root key
(3) keys in right subtree > root key
(4) left and right subtrees are also binary search trees
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Binary Search Trees(Cont’)
20
15
12
30
25
5
22
10
(a)
40
2
70
65
(b)
Figure 5.28 : Binary trees
Dept. of Computer Eng. & IT, Amirkabir University of Technology
60
80
(c)
Binary Search Trees(Cont’)
Searching Binary Search Tree
Recursive search by key value
definition of binary search tree is recursive
key(element) = x
x=root key : element=root
x<root key : search left subtree
x>root key : search right subtree
Complexity : O(h)
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Binary Search Trees(Cont’)
Insertion into Binary Search Tree
New element x
search x in the tree
success : x is in the tree
fail : insert x at the point the search terminated
30
30
5
40
2
5
80
(a) Insert 80
2
40
35
(b) Insert 35
Figure 5.29 : Inserting into a binary search tree
Dept. of Computer Eng. & IT, Amirkabir University of Technology
80
Binary Search Trees(Cont’)
template <class Type>
Boolean BST<Type>::Insert(const Element<Type> &x)
// Insert x into the binary search tree
{
// Search for x.key, q is parent of p
BstNode<Type> *p = root; BstNode<Type> *q = 0;
while(p) {
q = p;
if (x.key = = p->data.key) return FALSE;
// x.key is already in tree
if (x.key < p->data.key) p = p->LeftChild;
else p = p->RightChild;
}
// Perform insertion
p = new BstNode<Type>;
p->LeftChild = p->RightChild = 0; p->data = x;
if (!root) root = p;
else if (x.key < q->data.key) q->LeftChild = p;
else q->RightChild = p;
return TRUE;
}
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Binary Search Trees(Cont’)
Deletion from Binary Search Tree
Leaf node
corresponding child field of its parent is set to 0
the node is disposed
Nonleaf node with one child
the node is disposed
child takes the place of the node
Nonleaf node with two children
node is replaced by either
the largest node in its left subtree
the smallest node in its right subtree
delete the replacing node from the subtree
Dept. of Computer Eng. & IT, Amirkabir University of Technology
Binary Search Trees(Cont’)
Dept. of Computer Eng. & IT, Amirkabir University of Technology