Trees 2

1
Trees (2)

• Threaded Binary Search Trees
• Tree Balancing AVL Trees
• Other trees

2
Review of Traversals in Binary Search Tree

• Preorder
• Inorder
• Postorder
• Recursive
• Complexity

3
Links in a Binary Tree

• Consider using a linked structure to implement a
  binary tree.
• If a binary tree has n nodes, then the total
  number of links is 2n.
• Since each node except the root has exactly one
  incoming arc, there are only n-1 links point to
  nodes the remaining n1 links are null.

data
4
Threaded Binary Search Trees

• Since there are more than half of the links not
  used, it would be efficient if we can use them
  for efficient traversals
• Right-threaded BST use the unused right pointer
  to the parent (inorder successor)
• Application inorder traversal
• Observation The inorder successor of a node x
  with right-thread is its parent if x is a left
  child of its parent otherwise, it is the nearest
  ancestor of x that contains x in its left subtree.

5
Example
H
E
K
L
B
F
J
G
M
I
D
A
C
Nodes with no right child
6
Example continue Right pointer
H
E
K
L
B
F
J
G
M
I
D
A
null
C
Nodes with no right child
7
Inorder traversal on the threaded BST
H
E
K
L
B
F
J
G
M
I
D
A
null
C

• Where is the save?

8
Algorithm for inorder traversal on a threaded BST

• Initialize a pointer p to the root
• while p! null
• while p-gtleft ! null
• p p-gtleft
• visit p
• while p-gtright is a thread
• pp-gtright
• visit p
• pp-gtright

9
Homework

• Quiz 13.1 problem 3-6

10
Tree Balancing AVL Trees

• Motivation The order given for creating a BST
  determines the efficiency of the searching (the
  level), as shown in Ch. 10.
• Review of the tree created by different orders
  (Searching time O(level))
• O, E, T, C, U, M, P
• C, O, M, P, U, T, E
• C, E, M, O, P, T, U
• AVL keeping a BST balanced as items are inserted
  (Adelson-Velskii Landis)

11
Balance factor

• Defined for a node x
• Denote the height of the left subtree of x as l,
  the height of the right subtree of x as r
• The balance factor bl-r
• AVL is a BST in which the balance factor of each
  node is 0, 1, or -1

12
AVL examples with balance factors indicated
1
-1
0
0
1
-1
0
0
0
0
0
?
?
0
?
How about this?
?
?
Are these AVL trees?
0
0
13
How to create an AVL tree?

• Mechanism
• Simple right rotation (applied to cases where the
  insertion is on a left subtree of a left child of
  the nearest ancestor with balance 2)
• Simple left rotation (right subtree, right child,
  ancestor balance is -2)
• Left-right rotation (right subtree, left child,
  ancestor balance 2)
• Right-left rotation (left subtree, right child,
  ancestor balance -2)

14
Simple right rotation exp

• Insertion happens on a left subtree of a left
  child of the nearest ancestor with balance 2

2
After rotation, the balance factor becomes 0
1
1
0
insertion
2
0
0
15
Left-right rotation

• When insertion occurs on a right subtree of a
  left child of the nearest ancestor with balance
  2)

P

2
P
Left rotation first
G
R
O
R
-1
0
Right rotation next Including the ancestor with
balance 2
0
D
O
G
1
Note rotation has to maintain the BST feature
insertion
M
M
D
0
16
Left-Right rotation continue

P
O
O
R


P
G

G
Right rotation next Including the ancestor
R
M

M

D
D
17
Homework Exercise 13.2

• 6-10 (construct the BST using AVL insertion
  algorithm)

18
Other Trees

• Binary trees cannot represent all situations
• 2-3-4 Tree
• Each node stores at most 3 data values
• Each internal node is a 2-node, 3-node, or 4-node
• All the leaves are on the same level

19
2-3-4 Tree example
53
60 70
27 38
16 25
33 36
41 46 48
55 59
65 68
73 75 79
Search item 36
20
B-Trees

• A B-tree of order m
• Each node stores at most m-1 data values
• Each internal node is a 2-node, 3-node, or an
  m-node
• All the leaves are on the same level
• B-Trees are useful for organizing data stored in
  secondary memory

21
Tree Graph

• A tree is a connected acyclic graph
• An acyclic graph is one that contains no cycles
• Graph Theory with Applications, J.A. Bondy and
  U.S.R. Murty
• What is a graph ?