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Data Structures

Data Structures. Binary Trees. Azhar Maqsood School of Electrical Engineering and Computer Sciences (SEECS-NUST). Visiting and Traversing a Node. Many applications require that all of the nodes of a tree be “ visited ”.

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Data Structures

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  1. Data Structures Binary Trees Azhar Maqsood School of Electrical Engineering and Computer Sciences (SEECS-NUST)

  2. 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.

  3. 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.

  4. 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.

  5. Tree Traversal Types

  6. 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 subtreebefore the subsequent left and right subtrees.

  7. Preorder Traversal

  8. Preorder Traversal A B C D G E H J I K • Visit root before traversing subtrees. F

  9. 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.

  10. Inorder Traversal

  11. 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))

  12. Example of Binary Tree (inorder)

  13. Inorder Traversal A B C D G E H J I K • Visit root between left and right subtree. F

  14. y 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.

  15. Postorder traversal Postorder Traversal

  16. yyyyyyyy 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

  17. Postorder traversal A B C D G E H J I K • Visit root after traversing subtrees. F

  18. 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

  19. 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.

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