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# Chapter 12 PowerPoint PPT Presentation

Chapter 12. Binary search trees Lee, Hsiu-Hui Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the web. Binary Search Tree. Binary-search property :

Chapter 12

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## Chapter 12

Binary search trees

Lee, Hsiu-Hui

Ack: This presentation is based on the lecture slides from Hsu, Lih-Hsing, as well as various materials from the web.

### Binary Search Tree

• Binary-search property:

Let x be a node in a binary search tree. If y is a node in the left subtree of x, then key[y]  key[x]. If y is a node in the right subtree of x, then key[x]  key[y].

Hsiu-Hui Lee

Hsiu-Hui Lee

### Inorder tree walk

INORDER_TREE_WALK(x)

1if

2then INORDER_TREE_WALK(left[x])

3print key[x]

4INORDER_TREE_WALK(right[x])

Hsiu-Hui Lee

### Theorem 12.1

If x is the root of an n-node subtree, then the call INORDER-TREE-WALK(x) takes (n) time.

Proved by substitution method.

Hsiu-Hui Lee

• Preorder tree walk

• Postorder tree walk

Hsiu-Hui Lee

### Querying a binary search tree

Hsiu-Hui Lee

TREE_SEARCH(x, k)

1ifor

2then returnx

3if

4then return TREE_SEARCH(left[x],k)

5else return TREE_SEARCH(right[x],k)

Hsiu-Hui Lee

ITERATIVE_SEARCH (x, k)

1While or

2do if

3then

4then

5returnx

Hsiu-Hui Lee

### MAXIMUM and MINIMUM

TREE_MINIMUM(x)

1 while left[x]  NIL

2 dox  left[x]

• returnx

TREE_MAXIMUM(x)

1 while right[x]  NIL

2 dox  right[x]

3 returnx

Hsiu-Hui Lee

### SUCCESSOR and PREDECESSOR

TREE_SUCCESSOR

1 if

2 then return TREE_MINIMUM(right[x])

3

4 whileand

5 do

6

7 returny

Hsiu-Hui Lee

• Successor of the node with key value 15. (Answer: 17)

• Successor of the node with key value 6. (Answer: 7)

• Successor of the node with key value 4. (Answer: 6)

• Predecessor of the node with key value 6. (Answer: 4)

Hsiu-Hui Lee

### Theorem 12.2

• The dynamic-set operations, SEARCH, MINIMUM, MAXIMUM, SUCCESSOR, and PREDECESSOR can be made to run in O(h) time on a binary search tree of height h.

Hsiu-Hui Lee

## 12.3 Insertion and deletion

Tree-Insert(T, z)

1 y  NIL

2 x root[T]

3 whilex NIL

4 doy x

5ifkey[z] < key[x]

6 thenx left[x]

7 elsex right[x]

• p[z]  y

9 ify = NIL

10 thenroot[T]  z tree T was empty

11 elseifkey[z] < key[y]

12 thenleft[y]  z

13elseright[y]  z

Hsiu-Hui Lee

Inserting an item with key 13 into a binary search tree

Hsiu-Hui Lee

Tree-Delete(T, z)

1 ifleft[z] = NILorright[z] = NIL

2 theny z

3 elsey Tree-Successor(z)

4 if left[y]  NIL

5thenx left[y]

6elsex right[y]

7 ifx NIL

• thenp[x]  p[y]

9 ifp[y] = NIL

10 thenroot[T]  x

11else ify = left[p[y]]

12thenleft[p[y]]  x

13elseright[p[y]]  x

14 ify z

15thenkey[z]  key[y]

16 copy y’s satellite data into z

17 returny

Hsiu-Hui Lee

Hsiu-Hui Lee

Hsiu-Hui Lee

Hsiu-Hui Lee

### Theorem 12.3

• The dynamic-set operations, INSERT and DELETE can be made to run in O(h) time on a binary search tree of height h.

Hsiu-Hui Lee