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C++ Programming: Program Design Including Data Structures, Fourth Edition Chapter 20: Binary Trees

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Objectives In this chapter, you will: Learn about binary trees Explore various binary tree traversal algorithms Learn how to organize data in a binary search tree Learn how to insert and delete items in a binary search tree Explore nonrecursive binary tree traversal algorithms C++ Programming: Program Design Including Data Structures, Fourth Edition2

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Binary Trees C++ Programming: Program Design Including Data Structures, Fourth Edition3

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Binary Trees (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition4 Right child of ALeft child of A Root node, and Parent of B and C Directed edge, directed branch, or branch Node Empty subtree (F’s right subtree)

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Binary Trees (continued) Every node has at most two children A node: −Stores its own information −Keeps track of its left subtree and right subtree lLink and rLink pointers C++ Programming: Program Design Including Data Structures, Fourth Edition9

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Binary Trees (continued) A pointer to the root node of the binary tree is stored outside the tree in a pointer variable C++ Programming: Program Design Including Data Structures, Fourth Edition10

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Binary Trees (continued) Leaf: node that has no left and right children U is parent of V if there’s a branch from U to V There’s a unique path from root to every node Length of a path: number of branches on path Level of a node: number of branches on the path from the root to the node −The level of the root node of a binary tree is 0 Height of a binary tree: number of nodes on the longest path from the root to a leaf C++ Programming: Program Design Including Data Structures, Fourth Edition11

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A leaf A is the parent of B and C A B D G is a path (of length 3) from node A to node G The longest path from root to a leaf is A B D G I The number of nodes on this path is 5 the height of the tree is 5

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Binary Trees (continued) How can we calculate the height of a binary tree? This is a recursive algorithm: C++ Programming: Program Design Including Data Structures, Fourth Edition13

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Copy Tree C++ Programming: Program Design Including Data Structures, Fourth Edition14

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Binary Tree Traversal Inorder traversal −Traverse the left subtree −Visit the node −Traverse the right subtree Preorder traversal −Visit the node −Traverse the left subtree −Traverse the right subtree C++ Programming: Program Design Including Data Structures, Fourth Edition15

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Binary Tree Traversal (continued) Postorder traversal −Traverse the left subtree −Traverse the right subtree −Visit the node C++ Programming: Program Design Including Data Structures, Fourth Edition16

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Binary Tree Traversal (continued) Inorder sequence: listing of the nodes produced by the inorder traversal of the tree Preorder sequence: listing of the nodes produced by the preorder traversal of the tree Postorder sequence: listing of the nodes produced by the postorder traversal of the tree C++ Programming: Program Design Including Data Structures, Fourth Edition17

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Binary Tree Traversal (continued) Inorder sequence: B D A C Preorder sequence: A B D C Postorder sequence: D B C A C++ Programming: Program Design Including Data Structures, Fourth Edition18

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Implementing Binary Trees Typical operations: −Determine whether the binary tree is empty −Search the binary tree for a particular item −Insert an item in the binary tree −Delete an item from the binary tree −Find the height of the binary tree −Find the number of nodes in the binary tree −Find the number of leaves in the binary tree −Traverse the binary tree −Copy the binary tree C++ Programming: Program Design Including Data Structures, Fourth Edition20

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Binary Search Trees We can traverse the tree to determine whether 53 is in the binary tree this is slow C++ Programming: Program Design Including Data Structures, Fourth Edition27

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Binary Search Trees (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition28

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Binary Search Trees (continued) Every binary search tree is a binary tree C++ Programming: Program Design Including Data Structures, Fourth Edition29

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Binary Search Trees (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition30

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Search C++ Programming: Program Design Including Data Structures, Fourth Edition32

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Insert C++ Programming: Program Design Including Data Structures, Fourth Edition33

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Insert (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition34

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Delete C++ Programming: Program Design Including Data Structures, Fourth Edition35

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Delete (continued) The delete operation has four cases: −The node to be deleted is a leaf −The node to be deleted has no left subtree −The node to be deleted has no right subtree −The node to be deleted has nonempty left and right subtrees C++ Programming: Program Design Including Data Structures, Fourth Edition36

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Delete (continued) To delete an item from the binary search tree, we must do the following: −Find the node containing the item (if any) to be deleted −Delete the node C++ Programming: Program Design Including Data Structures, Fourth Edition42

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Binary Search Tree: Analysis Let T be a binary search tree with n nodes, where n > 0 Suppose that we want to determine whether an item, x, is in T The performance of the search algorithm depends on the shape of T In the worst case, T is linear C++ Programming: Program Design Including Data Structures, Fourth Edition47

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Binary Search Tree: Analysis (continued) Worst case behavior: T is linear –O(n) key comparisons C++ Programming: Program Design Including Data Structures, Fourth Edition48

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Binary Search Tree: Analysis (continued) Average-case behavior: −There are n! possible orderings of the keys We assume that orderings are possible −S(n) and U(n): number of comparisons in average successful and unsuccessful case, respectively C++ Programming: Program Design Including Data Structures, Fourth Edition49

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Binary Search Tree: Analysis (continued) C++ Programming: Program Design Including Data Structures, Fourth Edition50

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Nonrecursive Binary Tree Traversal Algorithms The traversal algorithms discussed earlier are recursive This section discusses the nonrecursive inorder, preorder, and postorder traversal algorithms C++ Programming: Program Design Including Data Structures, Fourth Edition51

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Nonrecursive Inorder Traversal For each node, the left subtree is visited first, then the node, and then the right subtree C++ Programming: Program Design Including Data Structures, Fourth Edition52 Will be visited first

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Nonrecursive Preorder Traversal For each node, first the node is visited, then the left subtree, and then the right subtree C++ Programming: Program Design Including Data Structures, Fourth Edition54

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Nonrecursive Postorder Traversal Visit order: left subtree, right subtree, node We must indicate to the node whether the left and right subtrees have been visited −Solution: other than saving a pointer to the node, save an integer value of 1 before moving to the left subtree and value of 2 before moving to the right subtree −When the stack is popped, the integer value associated with that pointer is popped as well C++ Programming: Program Design Including Data Structures, Fourth Edition55

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Binary Tree Traversal and Functions as Parameters In a traversal algorithm, “visiting” may mean different things −Example: output value, update value in some way Problem: How do we write a generic traversal function? −Writing a specific traversal function for each type of “visit” would be cumbersome Solution: pass a function as a parameter to the traversal function C++ Programming: Program Design Including Data Structures, Fourth Edition56

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Binary Tree Traversal and Functions as Parameters (continued) In C++, a function name without parentheses is considered a pointer to the function To specify a function as a formal parameter to another function: −Specify the function type, followed by name as a pointer, followed by the parameter types C++ Programming: Program Design Including Data Structures, Fourth Edition57

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Summary A binary tree is either empty or it has a special node called the root node −If the tree is nonempty, the root node has two sets of nodes (left and right subtrees), such that the left and right subtrees are also binary trees The node of a binary tree has two links in it A node in the binary tree is called a leaf if it has no left and right children A node U is called the parent of a node V if there is a branch from U to V C++ Programming: Program Design Including Data Structures, Fourth Edition60

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Summary (continued) Level of a node: number of branches on the path from the root to the node −The level of the root node of a binary tree is 0 −The level of the children of the root is 1 Height of a binary tree: number of nodes on the longest path from the root to a leaf C++ Programming: Program Design Including Data Structures, Fourth Edition61

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Summary (continued) Inorder traversal −Traverse left, visit node, traverse right Preorder traversal −Visit node, traverse left, traverse right Postorder traversal −Traverse left, traverse right, visit node C++ Programming: Program Design Including Data Structures, Fourth Edition62

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