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Data Structures – Binary Tree

Data Structures – Binary Tree. What is a tree?. Where do you see trees?. Ummm...outside file systems. Tree Terminology. node – any element of the tree root – the topmost node of the tree parent – a node that has one or more nodes connected below it (children)

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Data Structures – Binary Tree

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  1. Data Structures – Binary Tree

  2. What is a tree?

  3. Where do you see trees? • Ummm...outside • file systems

  4. Tree Terminology • node – any element of the tree • root – the topmost node of the tree • parent – a node that has one or more nodes connected below it (children) • child – a node that has a connected node above it (parent) • leaf – any child node at the bottom of the tree • subtree – a parent and all the nodes below it

  5. Name that part! root / parent subtree parent / child’ child / leaf child / leaf child / leaf

  6. So what’s a binary tree? • A parent can have at most TWO children (left child, right child) • The left child’s data and all the data in the left subtree is “less than” the parent’s data • The right child’s data all the data in the right subtree is “greater than” the parent’s data • NOTE: The “less than” and “greater than” requirements of the subtrees is commonly known outside of the IB world as a Binary Search Tree

  7. Example – Valid Binary Tree 6 2 10 4 1

  8. Example – INVALID Binary Tree 3 2 10 4 1

  9. Practice 1 • Create a binary tree by inserting the following numbers in the given order: 6, 4, 2, 7, 8, 14, 9

  10. Practice 2 • Create a binary tree by inserting the following numbers in the given order : 14, 75, 2, 34, 25, 26, 27, 28

  11. Another Binary Tree h d m a k x

  12. Practice 3 • Create a binary tree by inserting the following strings: Wanda, Alpos, Vazbyte, Fecso, Downs, Mata, Montante

  13. Practice 4 • Create a binary tree by inserting the following strings: Ramos, Dia, Khurelbaatar, Nice, Ren, Shahid, Zetkulic

  14. Adding to a Binary Tree • Start at the root • Compare against the current node • Go left if less than current node OR insert if there is none (creating a new leaf) • Go right if greater than current node OR insert if there is none (creating a new leaf) • Repeat step 2

  15. Searching in a Binary Tree • Start at the root • Compare against the current node • Found the node if they are equal • Go left if less than current node OR if there is no left, then does not exist • Go right if greater than current node OR if there is no left, then does not exist • Repeat step 2

  16. Search(4) – What path do you take? 6 2 10 4 13 1

  17. Search(9) – What path do you take? 6 2 10 4 13 1

  18. Removing from a Binary Tree • Search for matching node • If node is a leaf, then unlink its parent • If node is a parent of one child, then link the node’s parent to the node’s child • If node is a parent of two children, then travel down its right subtree to find the left-most leaf (smallest value of the right subtree). Take the value and put it in the original node that was being removed. Unlink the right-most leaf that you found.

  19. Remove(4) – 0 children case 6 2 10 4 13 1

  20. Remove(10) – 1 child case 6 2 10 4 13 1

  21. Remove(6) – 2 children case 6 2 10 4 13 1

  22. When do we use binary trees? • ...whenever we need to search for quickly • Inherent binary searching capabilities • Large trees can be searched quickly • What is the best case scenario? • What is the worst case scenario? • What is the average case scenario? • Compare a Linked List to a Binary Tree

  23. Tree Traversal • Add, remove, search only go down 1 path • How do you “walk-through” all the nodes of a tree?

  24. In-order tree traversal • Left-subtree traversal if it exists and rerun • Action on the current node (e.g. print) • Right-subtree traversal if it exists and rerun Note: In-order traversal often used to visit nodes in their inherent order

  25. In-order Traversal (1, 2, 4, 6, 8, 10, 13) 6 2 10 4 8 13 1

  26. Pre-order tree traversal • Action on the current node (e.g. copy) • Left-subtree traversal if it exists and rerun • Right-subtree traversal if it exists and rerun Note: Pre-order traversal is often used to duplicate a tree

  27. Pre-order Traversal (6, 2, 1, 4, 10, 8, 13) 6 2 10 4 8 13 1

  28. Post-order tree traversal • Left-subtree traversal if it exists and rerun • Right-subtree traversal if it exists and rerun • Action on the current node (e.g. print) Note: Post-order traversal is often used to completely delete or free up all nodes by visiting children and lowest levels first. (not really necessary with garbage collection)

  29. Post-order Traversal (1, 4, 2, 8, 13, 10, 6) 6 2 10 4 8 13 1

  30. Other Resources • http://www.csanimated.com/animation.php?t=Binary_search_tree

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