06 - Trees

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I used this set of slides for the lecture on Trees I gave at the University of Zurich for the 1st year students following the course of Formale Grundlagen der Informatik.

Technologie

n5
n2
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n1
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edge
node

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leaf
root

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n1
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parent
child

n5
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n1
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height = 1
depth = level = 2
degree = 1

n5
n2
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n1
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height = 1
depth = level = 1
degree = 3

n5
n2
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n10
n1
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n4
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ordered tree
leftmost rightmost

n5
n2
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n1
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n4
n9
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n8 n7
n4
n9n8
n1
n5
n4
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n3
n10
isomorphic trees

n4
n2
n5
n1
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n7
left right
left leftright right

n4
n2
n5
n1
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n6 n4
n2
n5
n1
n6
n3
not a binary tree binary tree
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n2
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n1
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n3
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complete binary tree

breadth-ﬁrst traversal
n4
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n3

breadth-ﬁrst traversal
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1

breadth-ﬁrst traversal
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2
1

breadth-ﬁrst traversal
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n1
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2 3
1

breadth-ﬁrst traversal
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2 3
4
1

breadth-ﬁrst traversal
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2 3
4 5
1

breadth-ﬁrst traversal
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2 3
4 65
1

n4
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n3
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
1
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
2
1
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
2
3
1
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
2
3 4
1
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
2 5
3 4
1
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
2 5
3 64
1
preorder(node)
visit(node).
preorder(node.left).
preorder(node.right).

n4
n2
n5
n1
n6
n3
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
n5
n1
n6
n3
1
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
n5
n1
n6
n3
2
1
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
n5
n1
n6
n3
2
1 3
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
n5
n1
n6
n3
2
1 3
4
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
n5
n1
n6
n3
2
1 53
4
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
n5
n1
n6
n3
2 6
1 53
4
inorder(node)
inorder(node.left).
visit(node).
inorder(node.right).

n4
n2
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n1
n6
n3
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

n4
n2
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n1
n6
n3
1
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

n4
n2
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n1
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n3
1 2
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

n4
n2
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n1
n6
n3
3
1 2
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

n4
n2
n5
n1
n6
n3
3
1 42
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

n4
n2
n5
n1
n6
n3
3 5
1 42
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

n4
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n1
n6
n3
3 5
1 42
6
postorder (node)
postorder(node.left).
postorder(node.right).
visit(node).

!"#$%&'(")$*
!
"#$#%&'!&$()*+,%&)$!&-!.,--#/!)$!(*)+!0#$#*,%&)$!%)!
Example (group 15)

Binary search tree
n4
n2
n5
n1
n6
n3
2 6
1 53
4

Inﬁx notation
a
+
b
-
c
*
d
Postﬁx notation
Preﬁx notation

Inﬁx notation
a
+
b
-
c
*
d
Postﬁx notation
Preﬁx notation -+ab*cd

Inﬁx notation
a
+
b
-
c
*
d
a+b-c*d
Postﬁx notation
Preﬁx notation -+ab*cd

Inﬁx notation
a
+
b
-
c
*
d
a+b-c*d
Postﬁx notation ab+cd*-
Preﬁx notation -+ab*cd

Tudor Gîrba
www.tudorgirba.com
creativecommons.org/licenses/by/3.0/

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06 - Trees

1. Trees www.tudorgirba.com

2. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8

3. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8

4. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8

5. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 edge node

6. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 leaf root

7. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 parent child

8. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 subtree

9. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 path

10. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 height = 1 depth = level = 2 degree = 1

11. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 height = 1 depth = level = 1 degree = 3

12. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 0 1 2 3

13. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 ordered tree leftmost rightmost

14. n5 n2 n6 n10 n1 n7 n4 n9 n3 n8 n7 n4 n9n8 n1 n5 n4 n6 n3 n10 isomorphic trees

15. n4 n2 n5 n1 n6 n3 n7 left right left leftright right

16. n4 n2 n5 n1 n3 n6 n4 n2 n5 n1 n6 n3 not a binary tree binary tree n4 n2 n5 n1 n6 n3 n7 complete binary tree

17. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3

18. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3 1

19. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3 2 1

20. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3 2 3 1

21. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3 2 3 4 1

22. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3 2 3 4 5 1

23. breadth-ﬁrst traversal n4 n2 n5 n1 n6 n3 2 3 4 65 1

24. n4 n2 n5 n1 n6 n3 preorder(node) visit(node). preorder(node.left). preorder(node.right).

25. n4 n2 n5 n1 n6 n3 1 preorder(node) visit(node). preorder(node.left). preorder(node.right).

26. n4 n2 n5 n1 n6 n3 2 1 preorder(node) visit(node). preorder(node.left). preorder(node.right).

27. n4 n2 n5 n1 n6 n3 2 3 1 preorder(node) visit(node). preorder(node.left). preorder(node.right).

28. n4 n2 n5 n1 n6 n3 2 3 4 1 preorder(node) visit(node). preorder(node.left). preorder(node.right).

29. n4 n2 n5 n1 n6 n3 2 5 3 4 1 preorder(node) visit(node). preorder(node.left). preorder(node.right).

30. n4 n2 n5 n1 n6 n3 2 5 3 64 1 preorder(node) visit(node). preorder(node.left). preorder(node.right).

31. n4 n2 n5 n1 n6 n3 inorder(node) inorder(node.left). visit(node). inorder(node.right).

32. n4 n2 n5 n1 n6 n3 1 inorder(node) inorder(node.left). visit(node). inorder(node.right).

33. n4 n2 n5 n1 n6 n3 2 1 inorder(node) inorder(node.left). visit(node). inorder(node.right).

34. n4 n2 n5 n1 n6 n3 2 1 3 inorder(node) inorder(node.left). visit(node). inorder(node.right).

35. n4 n2 n5 n1 n6 n3 2 1 3 4 inorder(node) inorder(node.left). visit(node). inorder(node.right).

36. n4 n2 n5 n1 n6 n3 2 1 53 4 inorder(node) inorder(node.left). visit(node). inorder(node.right).

37. n4 n2 n5 n1 n6 n3 2 6 1 53 4 inorder(node) inorder(node.left). visit(node). inorder(node.right).

38. n4 n2 n5 n1 n6 n3 postorder (node) postorder(node.left). postorder(node.right). visit(node).

39. n4 n2 n5 n1 n6 n3 1 postorder (node) postorder(node.left). postorder(node.right). visit(node).

40. n4 n2 n5 n1 n6 n3 1 2 postorder (node) postorder(node.left). postorder(node.right). visit(node).

41. n4 n2 n5 n1 n6 n3 3 1 2 postorder (node) postorder(node.left). postorder(node.right). visit(node).

42. n4 n2 n5 n1 n6 n3 3 1 42 postorder (node) postorder(node.left). postorder(node.right). visit(node).

43. n4 n2 n5 n1 n6 n3 3 5 1 42 postorder (node) postorder(node.left). postorder(node.right). visit(node).

44. n4 n2 n5 n1 n6 n3 3 5 1 42 6 postorder (node) postorder(node.left). postorder(node.right). visit(node).

45. !"#$%&'(")$* ! "#$#%&'!&$()*+,%&)$!&-!.,--#/!)$!(*)+!0#$#*,%&)$!%)! Example (group 15)

46. !"#$%&'(")$* ! "#$#%&'!&$()*+,%&)$!&-!.,--#/!)$!(*)+!0#$#*,%&)$!%)! Example (group 15)

47. Hierarchical structure

48. Binary search tree n4 n2 n5 n1 n6 n3 2 6 1 53 4

49. n4 n2 n5 n1 n6 n3 2 61 53 4

50. n4 n2 n5 n1 n6 n3 2 61 53 4

51. n4 n2 n5 n1 n6 n3 2 6 1 53 4

52. n4 n2 n5 n1 n6 n3 2 6 1 5 3 4

53. Binary search tree n4 n2 n5 n1 n6 n3 2 6 1 53 4

54. a + b - c * d

55. a + b - c * d

56. a + b - c * d

57. a + b - c * d operator operand

58. Infix notation a + b - c * d Postfix notation Prefix notation

59. Infix notation a + b - c * d Postfix notation Prefix notation -+ab*cd

60. Infix notation a + b - c * d a+b-c*d Postfix notation Prefix notation -+ab*cd

61. Infix notation a + b - c * d a+b-c*d Postfix notation ab+cd*- Prefix notation -+ab*cd

62. Tudor Gîrba www.tudorgirba.com creativecommons.org/licenses/by/3.0/

06 - Trees

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06 - Trees