Binary search trees Traversals (continued) Dictionary ADT Binary search tree February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 1
Announcements • Final exam: Monday, Apr.20, 15:30 – 18:00 – Location(s) TBA February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 2
Traversal Cost void preOrder(Node* nd) { void inOrder(Node* nd) { if (nd != NULL) { if (nd != NULL) { cout << nd->data << " "; inOrder(nd->left); preOrder(nd->left); cout << nd->data << " "; preOrder(nd->right); inOrder(nd->right); } } } } And post-order... • Cost? Let's try a recurrence relation: 𝑈 0 ≤ 𝑐 𝑈 𝑜 ≤ 𝑑 + 𝑈 𝑜 𝑀 + 𝑈 𝑜 𝑆 – Where 𝑜 𝑀 and 𝑜 𝑆 are the number of nodes in the left and right subtrees – But the shape/structure of the tree is not guaranteed! – Let's look at another property first February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 3
How many empty trees? • Consider an arbitrary binary tree with 𝑜 nodes – How many empty (null) subtrees are there? – Each of the 𝑜 nodes has 2 children (real or empty) • 2𝑜 "children" in total • 𝑜 − 1 of these are real nodes (root is not a child) • 𝑜 + 1 empty subtrees 𝑜 𝑆 • Proof (by strong induction): 𝑜 𝑀 𝐹 0 = 1 empty tree has single empty node 𝑜 𝑀 + 𝑜 𝑆 + 1 = 𝑜 𝐹 𝑜 = 𝐹 𝑜 𝑀 + 𝐹 𝑜 𝑆 = 𝑜 𝑀 + 1 + 𝑜 𝑆 + 1 by I.H. = 𝑜 𝑀 + 𝑜 𝑆 + 1 + 1 = 𝑜 + 1 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 4
Traversal Cost – A tree with 𝑜 nodes has 𝑜 + 1 empty nodes void preOrder(Node* nd) { if (nd != NULL) { – Pay 𝑑 at each of the 𝑜 real nodes cout << nd->data << " "; – Pay 𝑐 at each of the 𝑜 + 1 empty nodes preOrder(nd->left); – Cost: 𝑈 𝑜 ≤ 𝑑 ∙ 𝑜 + 𝑐 𝑜 + 1 preOrder(nd->right); } } • Proof (by strong induction): 𝑈 0 ≤ 𝑐 𝑈 𝑜 ≤ 𝑑 + 𝑈 𝑜 𝑀 + 𝑈 𝑜 𝑆 𝑈 𝑜 ≤ 𝑑 + 𝑑 ∙ 𝑜 𝑀 + 𝑐 𝑜 𝑀 + 1 + 𝑑 ∙ 𝑜 𝑆 + 𝑐 𝑜 𝑆 + 1 by I.H. 𝑈 𝑜 ≤ 𝑑 1 + 𝑜 𝑀 + 𝑜 𝑆 + 𝑐 𝑜 𝑀 + 𝑜 𝑆 + 1 + 1 𝑈 𝑜 ≤ 𝑑 ∙ 𝑜 + 𝑐 𝑜 + 1 ∈ 𝑃 𝑜 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 5
Traversal uses • Consider the following scenarios: – Copy constructor – Destructor – Which traversal (in-order, pre-order, post-order) would be most suitable? February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 6
Binary search tree Dictionary ADT Binary search tree February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 7
Motivation for an efficient lookup Dictionary ADT • Stores values associated with user-specified keys – Values may be any (homogenous) type – Keys may be any (homogenous) comparable type • Dictionary operations – Create • Super 9 LC Insert – Destroy • Smell like a lawnmower • – Insert Feet • • Useful for something, Z125 Pro – Find • presumably Fun in the sun! – Remove • CB300F Find(Z125 Pro) • For the mild-mannered commuter • Z125 Pro • Fun in the sun! February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 8
Data structures for Dictionary ADT Naïve implementations, complexity Search Insert Remove • Linked list • Unsorted array • Sorted array • Ordered linked list February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 9
Binary search tree property • A binary search tree is a binary tree with a special property – For all nodes in the tree: • All nodes in a left subtree have labels less than the label of the subtree's root • All nodes in a right subtree have labels greater than or equal to the label of the subtree's root • Binary search trees are fully ordered February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 10
BST example 17 13 27 9 16 20 39 11 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 11
BST inOrder traversal inOrder(nd->leftchild); inOrder traversal on a BST retrieves data in sorted order 5 visit(nd); inOrder(nd->rightchild); 17 3 7 inOrder(left) inOrder(left) 13 27 visit visit inOrder(right) inOrder(right) 8 4 6 1 inOrder(left) inOrder(left) inOrder(left) 16 20 39 visit 9 visit visit inOrder(right) inOrder(right) inOrder(right) inOrder(left) visit 2 inOrder(right) inOrder(left) 11 visit inOrder(right) February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 12
BST search • To find a value in a BST search from the root node: – If the target is less than the value in the node search its left subtree – If the target is greater than the value in the node search its right subtree – Otherwise return true, (or a pointer to the data, or …) • How many comparisons? – One for each node on the path – Worst case: height of the tree + 1 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 13
BST search example search(27); 17 27 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 14
BST search example search(16); 17 13 16 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 15
BST search example search(12); 17 13 9 11 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 16
BST insertion • The BST property must hold after insertion • Therefore the new node must be inserted in the correct position – This position is found by performing a search – If the search ends at the (null) left child of a node make its left child refer to the new node – If the search ends at the right child of a node make its right child refer to the new node • The cost is about the same as the cost for the search algorithm, O( height ) February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 17
BST insertion example Insert 43 47 Create new node Find position 32 63 Link node 19 41 54 79 43 10 23 37 44 53 59 96 7 12 30 43 57 91 97 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 18
BST insertion example Create new BST 3 Insert 3 Insert 15 15 Insert 21 Insert 23 21 Insert 37 Search 45 23 How many operations for Search? Complexity? 37 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 19
Find min, Find max • Find minimum: – From the root, keep following left child links until no more left child exists • Find maximum: – From the root, follow right child links until no more right child exists 43 18 68 12 33 52 7 27 39 50 56 9 21 67 February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 20
Readings for this lesson • Carrano & Henry – Chapter 15.2 – 15.3 (Binary/search tree) – Chapter 18.1 (Dictionary) • Next class: – Chapter 15.3, 16.1 – 16.2 (Binary search tree) February 12, 2020 Cinda Heeren / Andy Roth / Geoffrey Tien 21
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