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Trees: In these slides: Introduction to trees Applications of trees Tree traversal 2

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4 Introduction to Trees A tree is a connected undirected graph that contains no circuits. Theorem: There is a unique simple path between any two of its nodes.

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LOGO 5 Introduction to Trees A tree is a connected undirected graph that contains no circuits. Theorem: There is a unique simple path between any two of its nodes. A (not-necessarily-connected) undirected graph without simple circuits is called a forest. You can think of it as a set of trees having disjoint sets of nodes.

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LOGO 6 Introduction to Trees A tree is a connected undirected graph that contains no circuits. Theorem: There is a unique simple path between any two of its nodes. A (not-necessarily-connected) undirected graph without simple circuits is called a forest. You can think of it as a set of trees having disjoint sets of nodes. A leaf node in a tree or forest is any pendant or isolated vertex. An internal node is any non-leaf vertex (thus it has degree ≥ ? ).

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LOGO 7 Introduction to Trees A tree is a connected undirected graph that contains no circuits. Theorem: There is a unique simple path between any two of its nodes. A (not-necessarily-connected) undirected graph without simple circuits is called a forest. You can think of it as a set of trees having disjoint sets of nodes. A leaf node in a tree or forest is any pendant or isolated vertex. An internal node is any non-leaf vertex (thus it has degree ≥ 2 ).

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LOGO 8 Tree and Forest Examples A Tree: A Forest: Leaves in green, internal nodes in brown.

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LOGO 9 Rooted Trees A rooted tree is a tree in which one node has been designated the root. Every edge is (implicitly or explicitly) directed away from the root.

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LOGO 10 Rooted Trees A rooted tree is a tree in which one node has been designated the root. Every edge is (implicitly or explicitly) directed away from the root. Concepts related to rooted trees: Parent, child, siblings, ancestors, descendents, leaf, internal node, subtree.

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LOGO 11 Rooted Tree Examples Note that a given unrooted tree with n nodes yields n different rooted trees. root Same tree except for choice of root

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LOGO 12 Rooted-Tree Terminology Exercise Find the parent, children, siblings, ancestors, & descendants of node f. a b c d e f g h i jk l m n o p q r root

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LOGO 13 n-ary trees A rooted tree is called n-ary if every vertex has no more than n children. It is called full if every internal (non-leaf) vertex has exactly n children.

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LOGO 14 n-ary trees A rooted tree is called n-ary if every vertex has no more than n children. It is called full if every internal (non-leaf) vertex has exactly n children. A 2-ary tree is called a binary tree. These are handy for describing sequences of yes- no decisions. Example: Comparisons in binary search algorithm.

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LOGO 15 Which Tree is Binary? Theorem: A given rooted tree is a binary tree iff every node other than the root has degree ≤ ___, and the root has degree ≤ ___.

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LOGO 16 Ordered Rooted Tree This is just a rooted tree in which the children of each internal node are ordered.

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LOGO 17 Ordered Rooted Tree This is just a rooted tree in which the children of each internal node are ordered. In ordered binary trees, we can define: left child, right child left subtree, right subtree

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LOGO 18 Ordered Rooted Tree This is just a rooted tree in which the children of each internal node are ordered. In ordered binary trees, we can define: left child, right child left subtree, right subtree For n-ary trees with n>2, can use terms like “leftmost”, “rightmost,” etc.

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LOGO 19 Trees as Models Can use trees to model the following: Saturated hydrocarbons Organizational structures Computer file systems In each case, would you use a rooted or a non- rooted tree?

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LOGO 20 Some Tree Theorems Any tree with n nodes has e = n−1 edges. Proof: Consider removing leaves.

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LOGO 21 Some Tree Theorems Any tree with n nodes has e = n−1 edges. Proof: Consider removing leaves. A full m-ary tree with i internal nodes has n=mi+1 nodes, and =(m−1)i+1 leaves. Proof: There are mi children of internal nodes, plus the root. And, = n−i = (m−1)i+1. □

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LOGO 22 Some Tree Theorems Any tree with n nodes has e = n−1 edges. Proof: Consider removing leaves. A full m-ary tree with i internal nodes has n=mi+1 nodes, and =(m−1)i+1 leaves. Proof: There are mi children of internal nodes, plus the root. And, = n−i = (m−1)i+1. □ Thus, when m is known and the tree is full, we can compute all four of the values e, i, n, and, given any one of them.

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LOGO 23 Some More Tree Theorems Definition: The level of a node is the length of the simple path from the root to the node. The height of a tree is maximum node level. A rooted m-ary tree with height h is called balanced if all leaves are at levels h or h−1.

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LOGO 24 Some More Tree Theorems Definition: The level of a node is the length of the simple path from the root to the node. The height of a tree is maximum node level. A rooted m-ary tree with height h is called balanced if all leaves are at levels h or h−1. Theorem: There are at most m h leaves in an m-ary tree of height h. Corollary: An m-ary tree with leaves has height h≥ log m . If m is full and balanced then h= log m .

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LOGO 25

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LOGO 26 Applications of Trees Binary search trees A simple data structure for sorted lists

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LOGO 27 Applications of Trees Binary search trees A simple data structure for sorted lists Decision trees Minimum comparisons in sorting algorithms

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LOGO 28 Applications of Trees Binary search trees A simple data structure for sorted lists Decision trees Minimum comparisons in sorting algorithms Prefix codes Huffman coding

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LOGO 29 Applications of Trees Binary search trees A simple data structure for sorted lists Decision trees Minimum comparisons in sorting algorithms Prefix codes Huffman coding Game trees

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LOGO 30 Binary Search Trees A representation for sorted sets of items. Supports the following operations in Θ(log n) average-case time: Searching for an existing item. Inserting a new item, if not already present. Supports printing out all items in Θ(n) time. Note that inserting into a plain sequence a i would instead take Θ(n) worst- case time.

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LOGO 31 Binary Search Tree Format Items are stored at individual tree nodes. We arrange for the tree to always obey this invariant: For every item x, Every node in x’s left subtree is less than x. Every node in x’s right subtree is greater than x. 7 312 15915 02811 Example:

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LOGO 32 Recursive Binary Tree Insert procedure insert(T: binary tree, x: item) v := root[T] if v = null then begin root[T] := x; return “Done” end else if v = x return “Already present” else if x v} return insert(rightSubtree[T], x)

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LOGO 33 Decision Trees A decision tree represents a decision-making process. Each possible “decision point” or situation is represented by a node. Each possible choice that could be made at that decision point is represented by an edge to a child node.

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LOGO 34 Decision Trees A decision tree represents a decision-making process. Each possible “decision point” or situation is represented by a node. Each possible choice that could be made at that decision point is represented by an edge to a child node. In the extended decision trees used in decision analysis, we also include nodes that represent random events and their outcomes.

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LOGO 35 Coin-Weighing Problem Imagine you have 8 coins, one of which is a lighter counterfeit, and a free-beam balance. No scale of weight markings is required for this problem! How many weighings are needed to guarantee that the counterfeit coin will be found? ?

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LOGO 36 As a Decision-Tree Problem In each situation, we pick two disjoint and equal- size subsets of coins to put on the scale. The balance then “decides” whether to tip left, tip right, or stay balanced. A given sequence of weighings thus yields a decision tree with branching factor 3.

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LOGO 37 Applying the Tree Height Theorem The decision tree must have at least 8 leaf nodes, since there are 8 possible outcomes. In terms of which coin is the counterfeit one. Recall the tree-height theorem, h≥ log m . Thus the decision tree must have height h ≥ log 3 8 = 1.893… = 2. Let’s see if we solve the problem with only 2 weighings…

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LOGO 38 General Solution Strategy The problem is an example of searching for 1 unique particular item, from among a list of n otherwise identical items. Somewhat analogous to the adage of “searching for a needle in haystack.” Armed with our balance, we can attack the problem using a divide- and-conquer strategy, like what’s done in binary search. We want to narrow down the set of possible locations where the desired item (coin) could be found down from n to just 1, in a logarithmic fashion. Each weighing has 3 possible outcomes. Thus, we should use it to partition the search space into 3 pieces that are as close to equal-sized as possible. This strategy will lead to the minimum possible worst-case number of weighings required.

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LOGO 39 General Balance Strategy On each step, put n/3 of the n coins to be searched on each side of the scale. If the scale tips to the left, then: The lightweight fake is in the right set of n/3 ≈ n/3 coins. If the scale tips to the right, then: The lightweight fake is in the left set of n/3 ≈ n/3 coins. If the scale stays balanced, then: The fake is in the remaining set of n − 2 n/3 ≈ n/3 coins that were not weighed! Except if n mod 3 = 1 then we can do a little better by weighing n/3 of the coins on each side. You can prove that this strategy always leads to a balanced 3-ary tree.

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LOGO 40 Coin Balancing Decision Tree Here’s what the tree looks like in our case: 123 vs 456 1 vs. 2 left: 123 balanced: 78 right: 456 7 vs. 8 4 vs. 5 L:1 R:2B:3 L:4 R:5B:6 L:7 R:8

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LOGO 41

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LOGO Universal Address Systems To totally order the vertices of on ordered rooted tree: 1.Label the root with the integer 0. 2.Label its k children (at level 1) from left to right with 1, 2, 3, …, k. 3.For each vertex v at level n with label A, label its k v children, from left to right, with A.1, A.2, A.3, …, A.k v. 42

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LOGO Universal Address Systems To totally order the vertices of on ordered rooted tree: 1.Label the root with the integer 0. 2.Label its k children (at level 1) from left to right with 1, 2, 3, …, k. 3.For each vertex v at level n with label A, label its k v children, from left to right, with A.1, A.2, A.3, …, A.k v. This labeling is called the universal address system of the ordered rooted tree. 43

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LOGO Universal Address Systems 44

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LOGO Traversal Algorithms A traversal algorithm is a procedure for systematically visiting every vertex of an ordered rooted tree An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered 45

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LOGO Traversal Algorithms A traversal algorithm is a procedure for systematically visiting every vertex of an ordered rooted tree An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered The three common traversals are: Preorder traversal Inorder traversal Postorder traversal 46

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LOGO Traversal Let T be an ordered rooted tree with root r. Suppose T 1, T 2, …,T n are the subtrees at r from left to right in T. r T1T1 T2T2 TnTn 47

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LOGO Preorder Traversal r T1T1 T2T2 TnTn Step 1: Visit r Step 2: Visit T 1 in preorder Step 3: Visit T 2 in preorder. Step n+1: Visit T n in preorder 48

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LOGO Preorder Traversal r T1T1 T2T2 TnTn Step 1: Visit r Step 2: Visit T 1 in preorder Step 3: Visit T 2 in preorder. Step n+1: Visit T n in preorder 49

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LOGO Preorder Traversal r T1T1 T2T2 TnTn Step 1: Visit r Step 2: Visit T 1 in preorder Step 3: Visit T 2 in preorder. Step n+1: Visit T n in preorder 50

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LOGO Preorder Traversal r T1T1 T2T2 TnTn Step 1: Visit r Step 2: Visit T 1 in preorder Step 3: Visit T 2 in preorder. Step n+1: Visit T n in preorder 51

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LOGO Preorder Traversal r T1T1 T2T2 TnTn Step 1: Visit r Step 2: Visit T 1 in preorder Step 3: Visit T 2 in preorder. Step n+1: Visit T n in preorder 52

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LOGO Example M A R EY P M HJ QT Tree: Visiting sequence: 53

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LOGO Example AM A R EY P M HJ QT Tree: Visiting sequence: 54

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LOGO Example AMJ A R EY P M HJ QT Tree: Visiting sequence: 55

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LOGO Example AYMJ A R EY P M HJ QT Tree: Visiting sequence: 56

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LOGO Example ARYMJ A R EY P M HJ QT Tree: Visiting sequence: 57

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LOGO Example ARYMHJ A R EY P M HJ QT Tree: Visiting sequence: 58

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LOGO Example ARYPMHJ A R EY P M HJ QT Tree: Visiting sequence: 59

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LOGO Example ARYPMHJQ A R EY P M HJ QT Tree: Visiting sequence: 60

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LOGO Example ARYPMHJQT A R EY P M HJ QT Tree: Visiting sequence: 61

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LOGO Example AREYPMHJQT A R EY P M HJ QT Tree: Visiting sequence: 62

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LOGO The Preorder Traversal of T In which order does a preorder traversal visit the vertices in the ordered rooted tree T shown to the left? Remember: Visit root, then visit subtrees left to right. 63

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LOGO The Preorder Traversal of T 64

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LOGO The Preorder Traversal of T 65

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LOGO The Preorder Traversal of T 66

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LOGO The Preorder Traversal of T 67

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LOGO Inorder Traversal Step 1: Visit T 1 in inorder Step 2: Visit r Step 3: Visit T 2 in inorder. Step n+1: Visit T n in inorder r T1T1 T2T2 TnTn 68

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LOGO Inorder Traversal Step 1: Visit T 1 in inorder Step 2: Visit r Step 3: Visit T 2 in inorder. Step n+1: Visit T n in inorder r T1T1 T2T2 TnTn 69

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LOGO Inorder Traversal Step 1: Visit T 1 in inorder Step 2: Visit r Step 3: Visit T 2 in inorder. Step n+1: Visit T n in inorder r T1T1 T2T2 TnTn 70

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LOGO Inorder Traversal Step 1: Visit T 1 in inorder Step 2: Visit r Step 3: Visit T 2 in inorder. Step n+1: Visit T n in inorder r T1T1 T2T2 TnTn 71

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LOGO Inorder Traversal Step 1: Visit T 1 in inorder Step 2: Visit r Step 3: Visit T 2 in inorder. Step n+1: Visit T n in inorder r T1T1 T2T2 TnTn 72

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LOGO Example J A R EY P M HJ QT Tree: Visiting sequence: 73

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LOGO Example AJ A R EY P M HJ QT Tree: Visiting sequence: 74

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LOGO Example AMJ A R EY P M HJ QT Tree: Visiting sequence: 75

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LOGO Example ARMJ A R EY P M HJ QT Tree: Visiting sequence: 76

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LOGO Example ARYMJ A R EY P M HJ QT Tree: Visiting sequence: 77

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LOGO Example ARYPMJ A R EY P M HJ QT Tree: Visiting sequence: 78

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LOGO Example ARYPMHJ A R EY P M HJ QT Tree: Visiting sequence: 79

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LOGO Example ARYPMHJQ A R EY P M HJ QT Tree: Visiting sequence: 80

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LOGO Example ARYPMHJQT A R EY P M HJ QT Tree: Visiting sequence: 81

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LOGO Example AREYPMHJQT A R EY P M HJ QT Tree: Visiting sequence: 82

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LOGO The Inorder Traversal of T In which order does an inorder traversal visit the vertices in the ordered rooted tree T shown to the left? Remember: Visit leftmost tree, visit root, visit other subtrees left to right. 83

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LOGO The Inorder Traversal of T 84

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LOGO The Inorder Traversal of T 85

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LOGO The Inorder Traversal of T 86

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LOGO The Inorder Traversal of T 87

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LOGO Postorder Traversal Step 1: Visit T 1 in postorder Step 2: Visit T 2 in postorder. Step n: Visit T n in postorder Step n+1: Visit r r T1T1 T2T2 TnTn 88

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LOGO Postorder Traversal Step 1: Visit T 1 in postorder Step 2: Visit T 2 in postorder. Step n: Visit T n in postorder Step n+1: Visit r r T1T1 T2T2 TnTn 89

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LOGO Postorder Traversal Step 1: Visit T 1 in postorder Step 2: Visit T 2 in postorder. Step n: Visit T n in postorder Step n+1: Visit r r T1T1 T2T2 TnTn 90

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LOGO Postorder Traversal Step 1: Visit T 1 in postorder Step 2: Visit T 2 in postorder. Step n: Visit T n in postorder Step n+1: Visit r r T1T1 T2T2 TnTn 91

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LOGO Postorder Traversal Step 1: Visit T 1 in postorder Step 2: Visit T 2 in postorder. Step n: Visit T n in postorder Step n+1: Visit r r T1T1 T2T2 TnTn 92

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LOGO Example J A R EY P M HJ QT Tree: Visiting sequence: 93

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LOGO Example AJ A R EY P M HJ QT Tree: Visiting sequence: 94

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LOGO Example ARJ A R EY P M HJ QT Tree: Visiting sequence: 95

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LOGO Example ARPJ A R EY P M HJ QT Tree: Visiting sequence: 96

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LOGO Example ARPJQ A R EY P M HJ QT Tree: Visiting sequence: 97

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LOGO Example ARPJQT A R EY P M HJ QT Tree: Visiting sequence: 98

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LOGO Example ARPHJQT A R EY P M HJ QT Tree: Visiting sequence: 99

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LOGO Example ARYPHJQT A R EY P M HJ QT Tree: Visiting sequence: 100

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LOGO Example AREYPHJQT A R EY P M HJ QT Tree: Visiting sequence: 101

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LOGO Example AREYPMHJQT A R EY P M HJ QT Tree: Visiting sequence: 102

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LOGO The Postorder Traversal of T In which order does a postorder traversal visit the vertices in the ordered rooted tree T shown to the left? Remember: Visit subtrees left to right, then visit root. 103

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LOGO The Postorder Traversal of T 104

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LOGO The Postorder Traversal of T 105

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LOGO The Postorder Traversal of T 106

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LOGO The Postorder Traversal of T 107

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LOGO Representing Arithmetic Expressions Complicated arithmetic expressions can be represented by an ordered rooted tree Internal vertices represent operators Leaves represent operands 108

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LOGO Representing Arithmetic Expressions Complicated arithmetic expressions can be represented by an ordered rooted tree Internal vertices represent operators Leaves represent operands Build the tree bottom-up Construct smaller subtrees Incorporate the smaller subtrees as part of larger subtrees 109

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LOGO Example (x+y) 2 + (x-3)/(y+2) 110

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LOGO Example (x+y) 2 + (x-3)/(y+2) + x y 111

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LOGO Example (x+y) 2 + (x-3)/(y+2) + x y 2 112

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LOGO Example (x+y) 2 + (x-3)/(y+2) + x y 2 – x 3 113

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LOGO Example (x+y) 2 + (x-3)/(y+2) + x y 2 – x 3 + y 2 114

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LOGO Example (x+y) 2 + (x-3)/(y+2) + x y 2 – x 3 + y 2 / 115

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LOGO Example (x+y) 2 + (x-3)/(y+2) + x y 2 – x 3 + y 2 / + 116

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LOGO Evaluating Prefix Notation In an prefix expression, a binary operator precedes its two operands The expression is evaluated right-left Look for the first operator from the right Evaluate the operator with the two operands immediately to its right 117

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 118

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 119

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 120

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 121

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 122

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 123

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 124

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 125

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 126

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 127

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 + 2 1 128

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 + 2 1 129

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LOGO Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 + 2 1 3 130

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder + – + / + 2 x y x 3 y 2 131

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + – + / + 2 x y x 3 y 2 132

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x y + – + / + 2 x y x 3 y 2 133

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y + – + / + 2 x y x 3 y 2 134

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 + – + / + 2 x y x 3 y 2 135

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 + – + / + 2 x y x 3 y 2 136

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x + – + / + 2 x y x 3 y 2 137

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x 3 + – + / + 2 x y x 3 y 2 138

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x – 3 + – + / + 2 x y x 3 y 2 139

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x – 3 y + – + / + 2 x y x 3 y 2 140

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x – 3 y 2 + – + / + 2 x y x 3 y 2 141

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x – 3 y + 2 + – + / + 2 x y x 3 y 2 142

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 x – 3 / y + 2 + – + / + 2 x y x 3 y 2 143

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LOGO Postfix Notation (Reverse Polish) Traverse in postorder x + y 2 + x – 3 / y + 2 + – + / + 2 x y x 3 y 2 144

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LOGO In an postfix expression, a binary operator follows its two operands The expression is evaluated left-right Look for the first operator from the left Evaluate the operator with the two operands immediately to its left Evaluating Postfix Notation 145

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 146

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 147

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 148

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 149

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 150

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 151

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 152

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 153

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 154

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 155

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 2 1 + 156

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LOGO Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 2 1 + 157

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LOGO Example 3 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 2 1 + 158

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LOGO 159 Spanning Trees You will learn about the following topics in tree later in other courses: Spanning Trees Minimum Spanning Trees

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LOGO Good Luck

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