BinarySearchTree-bddicken
- 1. Binary Search Tree Benjamin Dicken University of Arizona
- 2. Table of contents 1. Background 2. Binary Search Tree 3. Search 4. Insert 5. Delete 6. Implementation 7. Summary 1
- 3. Background
- 4. Background Up to this point, we have seen: • Binary search in an array data structure • The concept of a binary tree • Recursion We are going to use all of these concepts when learning about binary search trees. 2
- 6. Binary Search Tree A Binary Search Tree (BST) has the following properties: 3
- 7. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure 3
- 8. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure – Each node of the tree has at most two children (a “left” child and a “right” child) 3
- 9. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure – Each node of the tree has at most two children (a “left” child and a “right” child) – Each node can legally have zero, one, or two children 3
- 10. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure – Each node of the tree has at most two children (a “left” child and a “right” child) – Each node can legally have zero, one, or two children • The Binary Search Tree Property must hold 3
- 11. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure – Each node of the tree has at most two children (a “left” child and a “right” child) – Each node can legally have zero, one, or two children • The Binary Search Tree Property must hold – If duplicates not allowed: “The key in each node must be greater than all keys stored in the left sub-tree, and less-than all keys in the right sub-tree.” 3
- 12. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure – Each node of the tree has at most two children (a “left” child and a “right” child) – Each node can legally have zero, one, or two children • The Binary Search Tree Property must hold – If duplicates not allowed: “The key in each node must be greater than all keys stored in the left sub-tree, and less-than all keys in the right sub-tree.” – If duplicates allowed: “The key in each node must be greater than all keys stored in the left sub-tree, and less-than or equal-to all keys in the right sub-tree.” 3
- 13. Binary Search Tree A Binary Search Tree (BST) has the following properties: • Is a binary tree data structure – Each node of the tree has at most two children (a “left” child and a “right” child) – Each node can legally have zero, one, or two children • The Binary Search Tree Property must hold – If duplicates not allowed: “The key in each node must be greater than all keys stored in the left sub-tree, and less-than all keys in the right sub-tree.” – If duplicates allowed: “The key in each node must be greater than all keys stored in the left sub-tree, and less-than or equal-to all keys in the right sub-tree.” – This guarantees that the tree is in sorted order. 3
- 14. Binary Search Tree • If the Binary Search Tree Property holds, this guarantees that the tree is in sorted order, thus making it a BST. • An in-order traversal of the tree will visit the nodes in order from lowest to highest. • Let’s take a look at a few trees and apply these tests to them. 4
- 15. Binary Search Tree Is the following a binary search tree? . 37 . . 21 . 11 24 . 55 . 48 62 5
- 16. Binary Search Tree Is the following a binary search tree? . 37 . . 20 . 13 24 55 . 56 . 100 6
- 17. Binary Search Tree Is the following a binary search tree? . 37 . . 20 . 21 24 . 55 . 18 62 7
- 18. Binary Search Tree Is the following a binary search tree? . 30 20 . . 25 23 8
- 19. Binary Search Tree Why use a BST? What are the advantages of a BST compared to some of the data structures we have already learned about, such as... • An array? • A sorted array? • A linked-list? We will revisit this after discussing how searching, insertion, and deletion works. 9
- 20. Search
- 21. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: 10
- 22. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C 10
- 23. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! 10
- 24. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! • Else if K > C 10
- 25. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! • Else if K > C • If has a right child, continue recursively on the right child 10
- 26. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! • Else if K > C • If has a right child, continue recursively on the right child • Else, search key not in tree 10
- 27. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! • Else if K > C • If has a right child, continue recursively on the right child • Else, search key not in tree • Else K < C 10
- 28. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! • Else if K > C • If has a right child, continue recursively on the right child • Else, search key not in tree • Else K < C • If has a left child, continue recursively on the left child 10
- 29. Search How do we search for a key K in a binary search tree? Beginning at the root of the tree, do the following operations: • Call the key of the current node C • If K == C, our search is complete! • Else if K > C • If has a right child, continue recursively on the right child • Else, search key not in tree • Else K < C • If has a left child, continue recursively on the left child • Else, search key not in tree 10
- 30. Search Exercise: search for 40, 91, 33 . 45 . . 10 . 7 . 34 . 30 40 . 70 . 56 80 . 90 11
- 31. Insert
- 32. Insert How do we insert a key K in a binary search tree? Starting from the root node: 12
- 33. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C 12
- 34. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root 12
- 35. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) 12
- 36. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) • Else if K > C 12
- 37. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) • Else if K > C • If has a right child, continue recursively on the right child 12
- 38. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) • Else if K > C • If has a right child, continue recursively on the right child • Else, add right child with key K 12
- 39. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) • Else if K > C • If has a right child, continue recursively on the right child • Else, add right child with key K • Else K < C 12
- 40. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) • Else if K > C • If has a right child, continue recursively on the right child • Else, add right child with key K • Else K < C • If has a left child, continue recursively on the left child 12
- 41. Insert How do we insert a key K in a binary search tree? Starting from the root node: • Call the key of the current node C • If the root node is null, insert a new node with key K as the root • Else if K == C, key already exists in tree, no need to insert (assuming no duplicates) • Else if K > C • If has a right child, continue recursively on the right child • Else, add right child with key K • Else K < C • If has a left child, continue recursively on the left child • Else, add left child with key K 12
- 46. Insert After inserting 50 19 . 50 17
- 48. Insert After inserting 75 19 . 50 . 75 19
- 49. Insert Insert 40 19 . 50 . 75 20
- 50. Insert After inserting 40 19 . . 50 . 40 75 21
- 51. Insert Insert 7 19 . . 50 . 40 75 22
- 52. Insert After inserting 7 . 19 . 7 . 50 . 40 75 23
- 53. Insert Insert 12 . 19 . 7 . 50 . 40 75 24
- 54. Insert After inserting 12 . 19 . 7 . 12 . 50 . 40 75 25
- 55. Insert Insert 2 . 19 . 7 . 12 . 50 . 40 75 26
- 56. Insert After inserting 2 . 19 . . 7 . 2 12 . 50 . 40 75 27
- 57. Delete
- 58. Delete How do we do deletion? 28
- 59. Delete How do we do deletion? • Traverse the tree, searching for the node with key K 28
- 60. Delete How do we do deletion? • Traverse the tree, searching for the node with key K • If no node with key K exists, nothing to do! 28
- 61. Delete How do we do deletion? • Traverse the tree, searching for the node with key K • If no node with key K exists, nothing to do! • Else, execute the recursive delete operation. 28
- 62. Delete What does the delete operation look like? 29
- 63. Delete What does the delete operation look like? • Call the node needing deletion C 29
- 64. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done 29
- 65. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done 29
- 66. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: 29
- 67. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. 29
- 68. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. – In order successor: The left-most node of the right sub-tree. 29
- 69. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. – In order successor: The left-most node of the right sub-tree. – In order predecessor: The right-most node of the left sub-tree. 29
- 70. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. – In order successor: The left-most node of the right sub-tree. – In order predecessor: The right-most node of the left sub-tree. – Copy the key from G into C 29
- 71. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. – In order successor: The left-most node of the right sub-tree. – In order predecessor: The right-most node of the left sub-tree. – Copy the key from G into C – Call the delete operation recursively on node G 29
- 72. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. – In order successor: The left-most node of the right sub-tree. – In order predecessor: The right-most node of the left sub-tree. – Copy the key from G into C – Call the delete operation recursively on node G 29
- 73. Delete What does the delete operation look like? • Call the node needing deletion C • If C has 0 children: just drop the reference to it, and we’re done • Else if C has 1 child: replace the parent of C’s reference to C with a reference to the single child, and we’re done • Else C has 2 children: – Find the in-order successor (or predecessor) of C. Call this node G. – In order successor: The left-most node of the right sub-tree. – In order predecessor: The right-most node of the left sub-tree. – Copy the key from G into C – Call the delete operation recursively on node G When doing recursion in the 2-children case, eventually a node will be reached that has either 0 or 1 children, in which case we do one of the base-case operations described above. Let’s look at an example. 29
- 74. Delete Exercise: delete 7 . 45 . . 10 . 7 . 34 . 30 40 . 70 . 52 80 . 90 30
- 75. Delete Exercise: delete 7 . 45 . . 10 . 7 . 34 . 30 40 . 70 . 52 80 . 90 31
- 76. Delete Exercise: delete 7 . 45 . . 10 . 7 . 34 . 30 40 . 70 . 52 80 . 90 32
- 77. Delete Exercise: delete 7 . 45 . . 10 . 7 . 34 . 30 40 . 70 . 52 80 . 90 33
- 78. Delete After deleting 7 . 45 . 10 . . 34 . 30 40 . 70 . 52 80 . 90 34
- 79. Delete Exercise: delete 10 . 45 . 10 . . 34 . 30 40 . 70 . 52 80 . 90 35
- 80. Delete Exercise: delete 10 . 45 . 10 . . 34 . 30 40 . 70 . 52 80 . 90 36
- 81. Delete Exercise: delete 10 . 45 . 10 . . 34 . 30 40 . 70 . 52 80 . 90 37
- 82. Delete After deleting 10 . 45 . . 34 . 30 40 . 70 . 52 80 . 90 38
- 83. Delete Exercise: delete 45 . 45 . . 34 . 30 40 . 70 . 52 80 . 90 39
- 84. Delete Exercise: delete 45 . 45 . . 34 . 30 40 . 70 . 52 80 . 90 40
- 85. Delete Exercise: delete 45 . 45 . . 34 . 30 40 . 70 . 52 80 . 90 41
- 86. Delete Exercise: delete 45 . 52 . . 34 . 30 40 . 70 . 52 80 . 90 42
- 87. Delete Exercise: delete 45 . 52 . . 34 . 30 40 . 70 . 52 80 . 90 43
- 88. Delete After deleting 45 . 52 . . 34 . 30 40 . 70 . 80 . 90 44
- 89. Implementation
- 91. Summary
- 92. Summary Benefits of BSTs compared to: • Sorted Array • Faster insert and delete • Similar search performance • Linked-List • Slower insert • Faster delete and search 46