1. Principles of Computing – UFCFA3-30-1
Week-11
Trees Data Structure
Instructor : Mazhar H Malik
Global College of Engineering and Technology

2. Trees
A tree data structure can be defined recursively (locally) as a collection of nodes (starting at a root node), where each node is a data structure consisting of a value, together with a list of references to nodes (the "children"), with the constraints that no reference is duplicated, and none points to the root.

3. Tree Traversal

5. Traversal Algorithms
Depth First Search
preorder
inorder
postorder

6. Preoder, Inorder, Postorder
In Preorder, the root
is visited before (pre)
the subtrees traversals
In Inorder, the root is
visited in-between left
and right subtree traversal
is visited after (pre)
Preorder Traversal:
Visit the root
Traverse left subtree
Traverse right subtree
Inorder Traversal:
Traverse left subtree
Visit the root
Traverse right subtree
Postorder Traversal:
Traverse left subtree
Traverse right subtree
Visit the root

7. PreOrder Traversal

8. Inorder Traversal

9. Postorder Traversal

10. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output:

11. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a

12. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {b, c, d}
output: a

13. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {b, c, d}
output: a

14. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b

15. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {e, f}
output: a b

16. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {e, f}
output: a b

17. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e

18. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {j, k}
output: a b e

19. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {j, k}
output: a b e

20. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j

21. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {}
output: a b e j

22. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {j, k}
output: a b e j

23. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j k

24. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {n, o, p}
output: a b e j k

25. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {n, o, p}
output: a b e j k

26. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j k n

27. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {}
output: a b e j k n

28. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {n, o, p}
output: a b e j k n

29. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j k n o

30. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {}
output: a b e j k n o

31. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {n, o, p}
output: a b e j k n o

32. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j k n o p

33. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {}
output: a b e j k n o p

34. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {n, o, p}
output: a b e j k n o p

35. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {j, k}
output: a b e j k n o p

36. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {e, f}
output: a b e j k n o p

37. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j k n o p f

38. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {}
output: a b e j k n o p f

39. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {e, f}
output: a b e j k n o p f

40. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
for each of {b, c, d}
output: a b e j k n o p f

41. In which order does a preorder traversal visit the vertices in this ordered rooted tree?
procedure preorder(T: ordered rooted tree)
r := root of T
list r
for each child c of r from left to right
begin
T(c) := subtree with c as its root
preorder(T(c))
end
output: a b e j k n o p f c d g l m h i

42. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
output:

43. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = ?
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
output:

44. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
output:

45. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = ?
s = {}
output:

46. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
output:

47. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = ?
s = {}
output:

48. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {}
output: j

49. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {}
r = j, l = ?
s = {}
output: j

50. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r =e, l = j
s = {}
output: j e

51. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
output: j e

52. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = ?
s = {}
output: j e

53. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {}
output: j e

54. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {}
r = n, l = ?
s = {}
output: j e n

55. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {}
output: j e n k

56. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {o,p}
output: j e n k

57. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {o,p}
r = o, l = ?
s = {}
output: j e n k o

58. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {o,p}
output: j e n k o

59. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {o,p}
r = k, l = ?
s = {}
output: j e n k o p

60. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
r = k, l = n
s = {o,p}
output: j e n k o p

61. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
r = e, l = j
s = {k}
output: j e n k o p

62. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {}
output: j e n k o p b

63. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {f}
output: j e n k o p b

64. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {f}
r = f, l = e
s = {}
output: j e n k o p b f

65. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
r = b, l = e
s = {f}
output: j e n k o p b f

66. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
output: j e n k o p b f

67. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
call
stack
r = a, l = b
s = {c,d}
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
output: j e n k o p b f

68. In which order does a inorder traversal visit the vertices in this ordered rooted tree?
procedure inorder(T: ordered rooted tree)
r := root of T
if r is a leaf then list r
else
begin
l:= first child of r from left to right
T(l) := subtree with l as its root
inorder(T(l))
list r
for each child c of r except for l left to right
T(c) := subtree with c as its root
preorder(T(c))
end
output: j e n k o p b f a c l g m d h i

69. Illustrations for Traversals
Assume: visiting a node
is printing its label
Preorder:
Inorder:
Postorder: 1 3 11 9 8 4 6 5 7 12 10

70. Illustrations for Traversals (Contd.)
Assume: visiting a node
is printing its data
Preorder:
Inorder:
Postorder: 6 15 8 2 3 7 11 10 14 12 20 27 22 30

71. a j b k c g m d l i h f e preorder: a j k m l b c g i h d f e
inorder: m k l j a b i g h c f d e
postorder: m l k j i h g f e d c b a