Data Structures

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Data Structures
Binary Trees

• Azhar Maqsood
• School of Electrical Engineering and Computer
  Sciences (SEECS-NUST)

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Visiting and Traversing a Node

• Many applications require that all of the nodes
  of a tree be visited.
• Visiting a node may mean printing contents,
  retrieving information, making a calculation,
  etc.
• Traverse To visit all the nodes in a tree in a
  systematic fashion.
• A traversal can pass through a node without
  visiting it at that moment.

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Depth First and Breadth First Traversal

• Depth-first traversals using the top-down view
  of the tree structure. Traverse the root, the
  left subtree and the right subtree.
• Breadth-first or level-order traversal visit
  nodes in order of increasing depth. For nodes of
  same depth visit in left-to-right order.

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Traversal strategies

• Preorder traversal
• Work at a node is performed before its children
  are processed.
• Postorder traversal
• Work at a node is performed after its children
  are processed.
• Inorder traversal
• For each node
• First left child is processed, then the work at
  the node is performed, and then the right child
  is processed.

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Tree Traversal Types
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Preorder traversal

• In the preorder traversal, the root node is
  processed first, followed by the left subtree and
  then the right subtree.

• Preorder root node of each subtree before the
  subsequent left and right subtrees.

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Preorder Traversal
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Preorder Traversal

• Visit root before traversing subtrees.

F
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Inorder Traversal

• In the inorder traversal, the left subtree is
  processed first, followed by the root node, and
  then the right subtree.

• Inorder root node in betweenthe left and right
  subtrees.

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Inorder Traversal
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Inorder Traversal

• In an inorder traversal a node is visited after
  its left subtree and before its right Subtree
• Application draw a binary tree or Arithmetic
  expression printing

((2 (a - 1)) (3 b))
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Example of Binary Tree (inorder)
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Inorder Traversal

• Visit root between left and right subtree.

F
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Postorder traversal

• In the postorder traversal, the left subtree is
  processed first, followed by the right subtree,
  and then the root node.

• Postorder root node after
• the left and right subtrees.

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Postorder traversal
Postorder Traversal
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Postorder traversal

• In a postorder traversal, a node is visited after
  its descendants
• Application compute space used by files in a
  directory and its subdirectories

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Postorder traversal

• Visit root after traversing subtrees.

F
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Expression Trees



a


g
b
c

f
d
e
Expression tree for ( a b c) ((d e f)
g)
There are three notations for a mathematical
expression 1) Infix notation ( a (b
c)) (((d e) f) c) 2) Postfix notation a b
c d e f g 3) Prefix notation a
b c d e f g
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Expression Tree traversals

• Depending on how we traverse the expression tree,
  we can produce one of these notations for the
  expression represented by the three.
• Inorder traversal produces infix notation.
• This is a overly parenthesized notation.
• Print out the operator, then print put the left
  subtree inside parentheses, and then print out
  the right subtree inside parentheses.
• Postorder traversal produces postfix notation.
• Print out the left subtree, then print out the
  right subtree, and then printout the operator.
• Preorder traversal produces prefix notation.
• Print out the operator, then print out the right
  subtree, and then print out the left subtree.