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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

Binary search Tree

Hsiu-Hui Lee

Inorder tree walk

INORDER_TREE_WALK(x)

1 if

2 then INORDER_TREE_WALK(left[x])

3 print key[x]

4 INORDER_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)

1 ifor

2 then returnx

3 if

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

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

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

Tree-Insert(T, z)

1 y NIL

2 x root[T]

3 whilex NIL

4 doy x

5 ifkey[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

13 elseright[y] z

Hsiu-Hui Lee

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

2 theny z

3 elsey Tree-Successor(z)

4 if left[y] NIL

5 thenx left[y]

6 elsex right[y]

7 ifx NIL

- thenp[x] p[y]
9 ifp[y] = NIL

10 thenroot[T] x

11 else ify = left[p[y]]

12 thenleft[p[y]] x

13 elseright[p[y]] x

14 ify z

15 thenkey[z] key[y]

16 copy y’s satellite data into z

17 returny

Hsiu-Hui Lee

Case a: z has no children

Hsiu-Hui Lee

Case b: z has only one child

Hsiu-Hui Lee

Case c: z has two children

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

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