Binary search trees - PowerPoint PPT Presentation

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Binary search trees

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Of course, all the leaves nodes have both NIL left field and NIL right field. 12/22/09 ... z. 12/22/09. 8. Tree traversals. 16. 6. 14. 10. 5. 7. 9. 2. 8 ... – PowerPoint PPT presentation

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Title: Binary search trees

1
Binary search trees

Binhai Zhu
Computer Science Department, Montana State
University

2
Definition

Binary search tree
(1) Each node, besides a key field and some
satellite data, contains left, right, and p
pointers that point to its left child, its right
child and its parent.
(2) The root is the only node whose parent
field is NIL. Of course, all the leaves nodes
have both NIL left field and NIL right field.

3
Example
root
10
7
14
16
9
5
leaf
8
6
2
leaf
leaf
leaf
4
Operations
Search
10
7
14
16
9
5
8
6
2
5
Operations
Search Minimum Maximum Predecessor Successor Inser
t Delete
10
7
14
16
9
5
8
6
2
6
Binary search tree property
10
7
14
16
9
5
8
6
2
For any node x, let y be a node in the left tree
of x, then keyy keyx. Likewise, if y is a
node in the right subtree of x then keyxkeyy.
7
Binary search tree property
10
x
7
14
16
9
5
z
y
8
6
2
For any node x, let y be a node in the left tree
of x, then keyy keyx. Likewise, if z is a
node in the right subtree of x then keyxkeyz.
8
Tree traversals
Inorder Preorder Postorder
10
7
14
16
9
5
8
6
2
9
Tree traversals
Inorder(node x) If x ? NIL Inorder(x?left)
print(x) Inorder(x?right)
10
7
14
16
9
5
8
6
2
10
Tree traversals
Inorder(node x) If x ? NIL Inorder(x?left)
print(x) Inorder(x?right)
10
7
14
16
9
5
8
6
2
2,5,6,7,8,9,10,14,16
11
Tree traversals
Inorder(node x) If x ? NIL Inorder(x?left)
print(x) Inorder(x?right)
10
7
14
16
9
5
8
6
2
2,5,6,7,8,9,10,14,16 (thats exactly the sorted
ordering!)
12
Tree traversals
Preorder(node x) If x ? NIL print(x)
Preorder(x?left) Preorder(x?right)
10
7
14
16
9
5
8
6
2
13
Tree traversals
Preorder(node x) If x ? NIL print(x)
Preorder(x?left) Preorder(x?right)
10
7
14
16
9
5
8
6
2
10,7,5,2,6,9,8,14,16
14
Tree traversals
Postorder(node x) If x ? NIL
Postorder(x?left) Postorder(x?right)
print(x)
10
7
14
16
9
5
8
6
2
15
Tree traversals
Postorder(node x) If x ? NIL
Postorder(x?left) Postorder(x?right)
print(x)
10
7
14
16
9
5
8
6
2
2,6,5,8,9,7,16,14,10
16
Tree traversals
What is the running time?
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7
14
16
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5
8
6
2
17
Minimum and Maximum
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7
14
16
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5
8
6
2
18
Minimum and Maximum
10
Minimum(node x) while x ? left ? NIL do x
? x?left return x
7
14
16
9
5
8
6
2
19
Minimum and Maximum
10

Minimum(node x)
while x ? left ? NIL
do x ? x?left
return x
Maximum(node x)
while x ? right ? NIL
do x ? x?right
return x

7
14
16
9
5
8
6
2
20
Search
x
10
7
14
16
9
5
k11?
8
6
2
k6
21
Search
Search(node x, k) if x NIL or k keyx
then return x if x lt keyx then return
Search(x?left,k) else return
Search(x?right,k)
x
10
7
14
16
9
5
8
6
2
k6
22
Search
Search(node x, k) if x NIL or k keyx
then return x if x lt keyx then return
Search(x?left,k) else return
Search(x?right,k) Iterative-Search(node x,k)
while x?NIL and k?keyx if k lt keyx
then x ? x?left else x ? x?right
return x
x
10
7
14
16
9
5
8
6
2
k6
23
Successor
The successor of x is the node with the smallest
key greater than keyx.
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7
14
x
16
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5
8
6
2
24
Successor
Successor(node x) if x?right ? NIL then
return Minimum(x?right) y ? x?p while y?NIL and
xy?right x ? y y ? y?p return y
10
7
14
16
9
5
x
8
6
2
25
Successor
Successor(node x) if x?right ? NIL then
return Minimum(x?right) y ? x?p while y?NIL and
xy?right x ? y y ? y?p return y
10
7
14
16
9
5
x
8
6
2
Search, Minimum, Maximum, Successor all run in
O(h) time, where h is the height of the
corresponding binary search tree.
26
Insertion
Insert a new node z with keyzv into a tree T.

10
7
14
16
9
5
9.5
8
6
2
z
27
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
x
10
7
14
16
9
5
9.5
8
6
2
z
28
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
y
x
10
7
14
16
9
5
9.5
8
6
2
z
29
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
y
10
7
14
x
16
9
5
9.5
8
6
2
z
30
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
10
y
7
14
x
16
9
5
9.5
8
6
2
z
31
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
10
y
7
14
x
16
9
5
9.5
8
6
2
z
32
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
10
7
14
y
16
9
5
x?NIL
9.5
8
6
2
z
33
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y
10
7
14
y
16
9
5
9.5
8
6
2
z
34
Insertion
Insert(T,z) y ? NIL x ? root(T) While x ? NIL
y ? x if keyz lt keyx then x
? x?left else x ? x?right z?p ? y If y
NIL then rootT?z else if keyz lt key y
then y?left ? z else y?right ? z
10
7
14
y
16
9
5
9.5
8
6
2
z
35
Deletion
10
z
z
7
14
16
9
5
8
6
2
z
36
Deletion
10
7
14
16
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5
8
6
2
z
37
Deletion
10
7
14
16
9
5
X
8
6
2
z
38
Deletion
10
z
7
14
16
9
5
8
6
2
39
Deletion
10
X
z
7
14
X
16
9
5
8
6
2
40
Deletion
10
z
7
14
16
9
5
8
6
2
41
Deletion
10
z
7
14
16
9
5
8
6
2
Find the successor of z
42
Deletion
10
z
8
14
16
9
5
8
6
2
Find the successor y of z Replace z with y Delete
y (careful, as y might have a right child)
43
Deletion