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- 1. Chapter 10 Trees 1
- 2. Definition of Tree• A tree is a set of linked nodes, such that there is one and only one path from a unique node (called the root node) to every other node in the tree.• A path exists from node A to node B if one can follow a chain of pointers to travel from node A to node B. 2
- 3. PathsA set of linked nodes D F A G E B There is one path from A to B C There is a path from D to B There is also a second path from D to B. 3
- 4. Paths (cont.) D FA G E B C There is no path from C to any other node. 4
- 5. Cycles• There is no cycle (circle of pointers) in a tree.• Any linked structure that has a cycle would have more than one path from the root node to another node. 5
- 6. Example of a Cycle D A B C EC→D→B→E→C 6
- 7. Tree Cannot Have a Cycle D A B C E2 paths exist from A to C:1. A → C2. A → C → D → B → E → C 7
- 8. Example of a Tree In a tree, everyroot pair of linked A nodes have a parent-child relationship (the C B D parent is closer to the root) G E F 8
- 9. Example of a Tree (cont.)root For example, C is a A parent of G C B D G E F 9
- 10. Example of a Tree (cont.)root A E and F are children of D C B D G E F 10
- 11. Example of a Tree (cont.) The root node is theroot A only node that has no parent. C B D G E F 11
- 12. Example of a Tree (cont.) Leaf nodes (orroot leaves for short) A have no children. C B D G E F 12
- 13. Subtrees A subtree is a part of a tree root that is a tree in itself A B CI K D E F J G H subtree 13
- 14. Binary Trees• A binary tree is a tree in which each node can only have up to two children… 14
- 15. NOT a Binary Tree root C has 3 child nodes. A B CI K D E F J G H 15
- 16. Example of a Binary Tree The links in a tree root are often called A edges B C I K E J G H 16
- 17. Levels root level 0 A level 1 B C level 2 I K E level 3 J G HThe level of a node is the number of edges in the pathfrom the root node to this node 17
- 18. Full Binary Tree root A B C D E F G H I J K L M N OIn a full binary tree, each node has two children except forthe nodes on the last level, which are leaf nodes 18
- 19. Complete Binary Trees• A complete binary tree is a binary tree that is either – a full binary tree – OR – a tree that would be a full binary tree but it is missing the rightmost nodes on the last level 19
- 20. NOT a Complete Binary Trees root A B C D E F GH I Missing non-rightmost nodes on the last level 20
- 21. Complete Binary Trees (cont.) root A B C D E F GH I J K L Missing rightmost nodes on the last level 21
- 22. Complete Binary Trees (cont.) A full binary tree is also a complete binary root A tree. B C D E F GH I J K L M N O 22
- 23. Binary Search Trees• A binary search tree is a binary tree that allows us to search for values that can be anywhere in the tree.• Usually, we search for a certain key value, and once we find the node that contains it, we retrieve the rest of the info at that node.• Therefore, we assume that all values searched for in a binary search tree are distinct. 23
- 24. Properties of Binary Search Trees• A binary search tree does not have to be a complete binary tree.• For any particular node, – the key in its left child (if any) is less than its key. – the key in its right child (if any) is greater than its key. 24
- 25. Binary Search Tree Node The implementationtemplate <typename T> of a binary searchBSTNode { tree usually just T info; maintains a single BSTNode<T> *left; pointer in the private BSTNode<T> *right; section called}; root, to point to the root node. 25
- 26. Inserting Nodes Into a BST root: NULL BST starts off emptyObjects that need to be inserted (only key values areshown):37, 2, 45, 48, 41, 29, 20, 30, 49, 7 26
- 27. Inserting Nodes Into a BST (cont.) root 3737, 2, 45, 48, 41, 29, 20, 30, 49, 7 27
- 28. Inserting Nodes Into a BST (cont.) root 37 2 < 37, so insert 2 on the left side of 372, 45, 48, 41, 29, 20, 30, 49, 7 28
- 29. Inserting Nodes Into a BST (cont.) root 37 22, 45, 48, 41, 29, 20, 30, 49, 7 29
- 30. Inserting Nodes Into a BST (cont.) root 37 2 45 > 37, so insert it at the right of 3745, 48, 41, 29, 20, 30, 49, 7 30
- 31. Inserting Nodes Into a BST (cont.) root 37 45 245, 48, 41, 29, 20, 30, 49, 7 31
- 32. Inserting Nodes Into a BST (cont.) root 37 45 2 When comparing, we always start at the root node 48 > 37, so look to the right48, 41, 29, 20, 30, 49, 7 32
- 33. Inserting Nodes Into a BST (cont.) root 37 45 2 This time, there is a node already to the right of the root node. We then compare 48 to this node 48 > 45, and 45 has no right child, so we insert 48 on the right of 4548, 41, 29, 20, 30, 49, 7 33
- 34. Inserting Nodes Into a BST (cont.) root 37 45 2 4848, 41, 29, 20, 30, 49, 7 34
- 35. Inserting Nodes Into a BST (cont.) root 37 45 2 41 > 37, so look to the right 48 41 < 45, so look to the left – there is no left child, so insert41, 29, 20, 30, 49, 7 35
- 36. Inserting Nodes Into a BST (cont.) root 37 45 2 48 4141, 29, 20, 30, 49, 7 36
- 37. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 < 37, left 41 29 > 2, right29, 20, 30, 49, 7 37
- 38. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 4129, 20, 30, 49, 7 38
- 39. Inserting Nodes Into a BST (cont.) root 37 45 2 48 20 < 37, left 29 41 20 > 2, right 20 < 29, left20, 30, 49, 7 39
- 40. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 2020, 30, 49, 7 40
- 41. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 30 < 37 20 30 > 2 30 > 2930, 49, 7 41
- 42. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 20 3030, 49, 7 42
- 43. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 49 > 37 49 > 45 20 30 49 > 4849, 7 43
- 44. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 20 30 4949, 7 44
- 45. Inserting Nodes Into a BST (cont.) root 37 45 2 487 < 37 29 417>27 < 297 < 20 20 30 497 45
- 46. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 20 30 49 77 46
- 47. Inserting Nodes Into a BST (cont.) root 37 45 2 48 29 41 All elements have 20 30 49 been inserted7 47
- 48. Searching for a Key in a BST root 37 45 2 48 29 41 Searching for a key in a BST uses 20 30 49 the same logic7 Key to search for: 29 48
- 49. Searching for a Key in a BST (cont.) root 37 45 2 4829 < 37 29 41 20 30 49 7 Key to search for: 29 49
- 50. Searching for a Key in a BST (cont.) root 37 45 2 4829 > 2 29 41 20 30 49 7 Key to search for: 29 50
- 51. Searching for a Key in a BST (cont.) root 37 45 2 4829 == 29 29 41FOUND IT! 20 30 49 7 Key to search for: 29 51
- 52. Searching for a Key in a BST (cont.) root 37 45 2 48 29 41 20 30 497 Key to search for: 3 52
- 53. Searching for a Key in a BST (cont.) root 37 45 23 < 373>2 483 < 29 29 413 < 203<7 20 30 49 7 Key to search for: 3 53
- 54. Searching for a Key in a BST (cont.) root 37 45 2When the child pointeryou want to follow isset to NULL, the key 48 29 41you are looking for isnot in the BST 20 30 49 7 Key to search for: 3 54
- 55. Deleting a BST Node• Deleting a node in a BST is a little tricky – it has to be deleted so that the resulting structure is still a BST with each node greater than its left child and less than its right child.• Deleting a node is handled differently depending on whether the node: – has no children – has one child – has two children 55
- 56. Deletion Case 1: No Childrenroot 37 452 48 29 41 Node 49 has no children – to 20 30 49 delete it, we just remove it 56
- 57. Deletion Case 1: No Children (cont.)root 37 452 48 29 41 20 30 57
- 58. Deletion Case 2: One Childroot 37 452 48 29 41 Node 48 has one child – to delete 20 30 it, we just splice 49 it out 58
- 59. Deletion Case 2: One Child (cont.)root 37 452 48 29 41 Node 48 has one child – to delete 20 30 it, we just splice 49 it out 59
- 60. Deletion Case 2: One Child (cont.)root 37 452 29 41 20 30 49 60
- 61. Deletion Case 2: One Child (cont.)root 37 452 48 29 41 Another example: node 2 has one child 20 30 – to delete it we also 49 splice it out 61
- 62. Deletion Case 2: One Child (cont.)root 37 452 48 29 41 Another example: node 2 has one child 20 30 – to delete it we also 49 splice it out 62
- 63. Deletion Case 2: One Child (cont.)root 37 45 48 29 41 20 30 49 63
- 64. Deletion Case 3: Two Childrenroot 37 452 48 29 41 Node 37 has two children… 20 30 49 64
- 65. Deletion Case 3: Two Children (cont.)root 37 452 48 29 41 to delete it, first we find the greatest 20 30 node in its left 49 subtree 65
- 66. Deletion Case 3: Two Children (cont.) root 37 45 2First, we goto the left 48 29 41once, thenfollow theright pointersas far as we 20 30 49can 66
- 67. Deletion Case 3: Two Children (cont.)root 37 452 48 29 41 30 is the greatest 20 30 node in the left 49 subtree of node 37 67
- 68. Deletion Case 3: Two Children (cont.)root 37 452 48 29 41 Next, we copy the 20 30 object at node 30 49 into node 37 68
- 69. Deletion Case 3: Two Children (cont.)root 30 452 48 29 41 Finally, we delete the lower red node 20 using case 1 or 49 case 2 deletion 69
- 70. Deletion Case 3: Two Children (cont.)root 30 452 48 29 41 Let’s delete node 30 20 now 49 70
- 71. Deletion Case 3: Two Children (cont.)root 30 452 48 29 41 29 is the greatest node in the left subtree of 20 49 node 30 71
- 72. Deletion Case 3: Two Children (cont.)root 30 452 48 29 41 Copy the object at node 29 into node 30 20 49 72
- 73. Deletion Case 3: Two Children (cont.)root 29 452 48 29 41 This time, the lower red node has a child – to delete 20 49 it we use case 2 deletion 73
- 74. Deletion Case 3: Two Children (cont.)root 29 452 48 29 41 This time, the lower red node has a child – to delete 20 49 it we use case 2 deletion 74
- 75. Deletion Case 3: Two Children (cont.)root 29 452 48 41 20 49 75
- 76. Traversing a BST• There are 3 ways to traversal a BST (visit every node in BST):• 1. Preorder (parent → left → right)• 2. Inorder (left → parent → right)• 3. Postorder (left → right → parent) 76
- 77. Binary Search Tree Class Template1 template <typename T>2 class BinarySearchTree {3 BSTNode *root;4 void preOrderInternal (BST<T> *parent) { … }5 void inOrderInternal (BST<T> *parent) { … }6 void postOrderInternal (BST<T> *parent) { … }7 void insertInternal (BST<T> *parent,8 const T& newElement) { … }9 bool searchInternal (const T& target,10 T& foundElement) { … } 77
- 78. Binary Search Tree Class Template (cont.)11 public:12 BinarySearchTree() { … }13 ~BinarySearchTree() { … }14 bool isEmpty() { … }15 void preOrderTraversal() { … }16 void inOrderTraversal() { … }17 void postOrderTraversal() { … }18 void insert (T element) { … }19 bool search (T element, T& foundElement) { … }20 void makeEmpty() { … }21 } 78
- 79. Constructor, Destructor, IsEmpty()1 BinarySearchTree()2 : root(NULL) { }34 ~BinarySearchTree()5 {6 makeEmpty();7 }89 bool isEmpty()10 {11 return root == NULL:12 } 79
- 80. Printing Elements In Order2 void inOrderTraversal() The client uses the3 { driver called4 inOrderInternal (root); InOrderTraversal.5 } The recursive function inOrderInternal is called. inOrderInternal prints all element in BST in sorted order, starting from root node. 80
- 81. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) {5 inOrderInternal (parent->left);6 cout << parent->info << " "; The base case7 inOrderInternal (parent->right); occurs when8 } parent is NULL –9 } just returns. 81
- 82. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) { There are 25 inOrderInternal (parent->left); recursive calls under the if – the6 cout << parent->info << " "; first advances7 inOrderInternal (parent->right); the pointer to the8 } left child…9 } and the second advances the pointer to the right child. 82
- 83. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) { Each recursive5 inOrderInternal (parent->left); call approaches6 cout << parent->info << " "; the base case…7 inOrderInternal (parent->right); it goes down one8 } level through the9 } tree, and it get closer to the case where parent->left or parent->right is NULL. 83
- 84. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 { If the BST contains4 if (parent != NULL) { only one node, the5 inOrderInternal (parent->left); root node is the only6 cout << parent->info << " "; node – the left and7 inOrderInternal (parent->right); right pointers of the8 } root node will be set9 } to NULL. 84
- 85. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) { parent->left is5 inOrderInternal (parent->left); NULL, so NULL is passed into6 cout << parent->info << " "; parent of the new7 inOrderInternal (parent->right); inOrderInternal8 } function.9 } 85
- 86. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 { Since parent is4 if (parent != NULL) { NULL in the new5 inOrderInternal (parent->left); inOrderInternal6 cout << parent->info << " "; function, it will7 inOrderInternal (parent->right); return right away.8 }9 } 86
- 87. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) { parent->right is5 inOrderInternal (parent->left); NULL – thus,6 cout << parent->info << " "; the recursive7 inOrderInternal (parent->right); function call immediately8 } returns as9 } before. 87
- 88. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) {5 inOrderInternal (parent->left);6 cout << parent->info << " ";7 inOrderInternal (parent->right);8 }9 } In fact inOrderInternal works correctly when there is only 0, 1 or many node in the BST. 88
- 89. Printing Elements In Order (cont.)2 void inOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) {5 inOrderInternal (parent->left);6 cout << parent->info << " ";7 inOrderInternal (parent->right);8 }9 } inOrderInternal is called once from the driver inOrderTraversal. Then, for each node in the tree, 2 calls to inOrderInternal are made, giving us a total of 2n + 1 calls where n is the number of nodes. 89
- 90. Printing Elements Pre Order2 void preOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) {5 cout << parent->info << " ";6 preOrderInternal (parent->left);7 preOrderInternal (parent->right);8 }9 } preOrderInternal prints the parent first. 90
- 91. Printing Elements Post Order2 void postOrderInternal (BSTNode<T> *parent)3 {4 if (parent != NULL) {5 postOrderInternal (parent->left);6 postOrderInternal (parent->right);7 cout << parent->info << " ";8 }9 } postOrderInternal prints the parent last. 91
- 92. Searching and Insertion in BST1 bool search (T target, T& foundElement) {2 return searchInternal (root, target, foundElement);3 }45 void insert (const T& newElement) {6 insertInternal (root, newElement);7 } 92
- 93. Searching and Insertion in BST• We can actually figure out searchInternal and insertInternal using the same recursive logic as traversal in BST. 93
- 94. References• Childs, J. S. (2008). Trees. C++ Classes and Data Structures. Prentice Hall. 94

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