1. Binhai Zhu Computer Science Department, Montana State University
Binary search trees
Binhai Zhu
Computer Science Department, Montana State University
Frequently, presenters must deliver material of a technical nature to an audience unfamiliar with the topic or vocabulary. The material may be complex or heavy with detail. To present technical material effectively, use the following guidelines from Dale Carnegie Training®.
Consider the amount of time available and prepare to organize your material. Narrow your topic. Divide your presentation into clear segments. Follow a logical progression. Maintain your focus throughout. Close the presentation with a summary, repetition of the key steps, or a logical conclusion.
Keep your audience in mind at all times. For example, be sure data is clear and information is relevant. Keep the level of detail and vocabulary appropriate for the audience. Use visuals to support key points or steps. Keep alert to the needs of your listeners, and you will have a more receptive audience.
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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.
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3. Example
root 10 7 14 9 16 5
leaf 2 6 8
leaf
leaf
leaf
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4. Operations
Search 10 7 14 9 16 5 2 6 8
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5. Operations Search Minimum Maximum Predecessor Successor Insert Delete10 7 14 9 16 5 2 6 8
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6. Binary search tree property10 7 14 9 16 5 2 6 8
For any node x, let y be a node in the left tree of x, then key[y] ≤ key[x]. Likewise, if y is a node in the right subtree of x then key[x]≤key[y].
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7. Binary search tree property10 x 7 14 9 16 5 z y 2 6 8
For any node x, let y be a node in the left tree of x, then key[y] ≤ key[x]. Likewise, if z is a node in the right subtree of x then key[x]≤key[z].
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8. Tree traversals Inorder Preorder Postorder 10 7 14 9 16 5 2 6 8
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9. Tree traversals Inorder(node x) If x ≠ NIL Inorder(x→left) print(x)
Inorder(x→right) 10 7 14 9 16 5 2 6 8
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10. Tree traversals Inorder(node x) If x ≠ NIL Inorder(x→left) print(x)
Inorder(x→right) 10 7 14 9 16 5 2 6 8
2,5,6,7,8,9,10,14,16
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11. Tree traversals Inorder(node x) If x ≠ NIL Inorder(x→left) print(x)
Inorder(x→right) 10 7 14 9 16 5 2 6 8
2,5,6,7,8,9,10,14,16 (that’s exactly the sorted ordering!)
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12. Tree traversals Preorder(node x) If x ≠ NIL print(x) Preorder(x→left)
Preorder(x→right) 10 7 14 9 16 5 2 6 8
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13. Tree traversals Preorder(node x) If x ≠ NIL print(x) Preorder(x→left)
Preorder(x→right) 10 7 14 9 16 5 2 6 8
10,7,5,2,6,9,8,14,16
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14. Tree traversals Postorder(node x) If x ≠ NIL Postorder(x→left)
Postorder(x→right)
print(x) 10 7 14 9 16 5 2 6 8
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15. Tree traversals Postorder(node x) If x ≠ NIL Postorder(x→left)
Postorder(x→right)
print(x) 10 7 14 9 16 5 2 6 8
2,6,5,8,9,7,16,14,10
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16. Tree traversals What is the running time? 10 7 14 9 16 5 2 6 8
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17. Minimum and Maximum 10 7 14 9 16 5 2 6 8
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18. Minimum and Maximum 10 Minimum(node x) while x → left ≠ NIL
do x ← x→left
return x 7 14 9 16 5 2 6 8
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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 7 14 9 16 5 2 6 8
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20. Search x 10 7 14 9 16 5
k=11? 2 6 8
k=6
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21. Search Search(node x, k) if x = NIL or k =key[x] then return x
if x < key[x]
then return Search(x→left,k)
else return Search(x→right,k) x 10 7 14 9 16 5 2 6 8
k=6
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22. Search Search(node x, k) if x = NIL or k =key[x] then return x
if x < key[x]
then return Search(x→left,k)
else return Search(x→right,k)
Iterative-Search(node x,k)
while x≠NIL and k≠key[x]
if k < key[x]
then x ← x→left
else x ← x→right
return x x 10 7 14 9 16 5 2 6 8
k=6
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23. Successor
The successor of x is the node with the smallest key greater than key[x]. 10 7 x 14 9 16 5 2 6 8
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24. Successor Successor(node x) if x→right ≠ NIL
then return Minimum(x→right)
y ← x→p
while y≠NIL and x==y→right
x ← y
y ← y→p
return y
Successor 10 7 14 9 16 5 x 2 6 8
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25. Successor Successor(node x) if x→right ≠ NIL
then return Minimum(x→right)
y ← x→p
while y≠NIL and x==y→right
x ← y
y ← y→p
return y
Successor 10 7 14 9 16 5 x 2 6 8
Search, Minimum, Maximum, Successor all run in O(h) time, where h is the height of the corresponding binary search tree.
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26. Insertion Insert a new node z with key[z]=v into a tree T. 10 7 14 916 5 2 6 8
9.5 z
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27. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y x 10 7 14 9 16 5
9.5 2 6 8 z
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28. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y y x 10 7 14 9 16 5 2 6 8
9.5 z
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29. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y y 10 7 x 14 9 16 5 2 6 8
9.5 z
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30. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y 10 y 7 x 14 9 16 5 2 6 8
9.5 z
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31. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y 10 y 7 14 9 x 16 5 2 6 8
9.5 z
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32. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y 10 7 14 y 9 16 5
x←NIL
9.5 2 6 8 z
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33. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y 10 7 14 y 9 16 5
9.5 2 6 8 z
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34. Insertion Insert(T,z) y ← NIL x ← root(T) While x ≠ NIL y ← x
if key[z] < key[x]
then x ← x→left
else x ← x→right
z→p ← y
If y = NIL then root[T]←z
else if key[z] < key [y]
then y→left ← z
else y→right ← z 10 7 14 y 9 16 5
9.5 2 6 8 z
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35. Deletion 10 z z 7 14 9 16 5 2 6 8 z
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36. Deletion 10 7 14 9 16 5 2 6 8 z
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37. Deletion 10 7 14 9 16 5 X 2 6 8 z
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38. Deletion 10 z 7 14 9 16 5 2 6 8
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39. Deletion 10 X z 7 14 X 9 16 5 2 6 8
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40. Deletion 10 z 7 14 9 16 5 2 6 8
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41. Deletion 10 z 7 14 9 16 5 2 6 8
Find the successor of z
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42. Deletion 10 z 8 14 9 16 5 2 6 8 Find the successor y of z
Replace z with y
Delete y (careful, as y might have a right child)
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43. 1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 7 z 14 9 16 5 2 6 8 z
8.5
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44. 1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 7 z 14 y 9 16 5 2 6 8 y z y
8.5
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45. Deletion 10 z 7 z 14 y 9 x 16 5 2 6 8 y z y x 8.5 x=NIL
1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 7 z 14 y 9 x 16 5 2 6 8 y z y x
8.5
x=NIL
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46. Deletion 10 z 7 z 14 y 9 x 16 5 2 6 8 y z y x 8.5 x=NIL
1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 7 z 14 y 9 x 16 5 2 6 8 y z y x
8.5
x=NIL
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47. 1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 X z 7 z 14 y X 9 x 16 5 X 2 6 8 y
8.5 x
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48. 1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 7 z 14 y 9 x 16 5 2 6 8 y
8.5 x
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49. 1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 8 z 14 y 9 x 16 5 2 6 8 y
8.5 x
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50. 1. if z→left=NIL or z→right=NIL
then y ← z
else y ← Successor(z)
4. if y→left≠NIL
then x ← y→left
else x ← y→right
7. if x ≠ NIL
then x→p ← y→p
9. if y→p = NIL
10. then root[T] ← x
else if y = (y→p)→left
then (y→p)→left ← x
else (y→p)→right ← x
14.if y≠z
then key[z] ← key[y]
copy y’s data into z
17.return y
Deletion 10 z 8 z y 9 x 16 5 2 6 y
8.5 x
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