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Unformatted text preview: What is a (General) Tree? •  A (general) tree is a set of nodes with the following proper4es: –  The set can be empty. –  Otherwise, the set is par44oned into k+1 disjoint subsets: •  a tree consists of a dis4nguished node r, called root, and zero or more nonempty sub ­trees T1, T2, … , Tk, each of whose roots are connected by an edge from r. •  T is a tree if either –  T has no nodes, or –  T is of the form: r T1 T2 Tk where r is a node and T1, T2, ..., Tk are trees. 3/7/11 CS202  ­ Fundamentals of Computer Science II 1 What is a (General) Tree? (cont.) •  The root of each sub ­tree is said to be child of r, and r is the parent of each sub ­tree root. •  If a tree is a collecFon of N nodes, then it has N ­1 edges. •  A path from node n1 to nk is defined as a sequence of nodes n1,n2, …,nk such that ni is parent of ni+1 (1 ≤ i < k) –  There is a path from every node to itself. –  There is exactly one path from the root to each node. 3/7/11 CS202  ­ Fundamentals of Computer Science II 2 Tree Terminology Parent – The parent of node n is the node directly above in the tree. Child – The child of node n is the node directly below in the tree. •  If node m is the parent of node n, node n is the child of node m. Root – The only node in the tree with no parent. Leaf – A node with no children. Siblings – Nodes with a common parent. Ancestor – An ancestor of node n is a node on the path from the root to n. Descendant – A descendant of node n is a node on the path from n to a leaf. Subtree – A subtree of node n is a tree that consists of a child (if any) of n and the child’s descendants (a tree which is rooted by a child of node n) 3/7/11 CS202  ­ Fundamentals of Computer Science II 3 A Tree – Example A B C D H E I F J P K L G M N Q – Node A has 6 children: B, C, D, E, F, G. – B, C, H, I, P, Q , K, L, M, N are leaves in the tree above. – K, L, M are siblings since F is parent of all of them. 3/7/11 CS202  ­ Fundamentals of Computer Science II 4 Level of a node Level – The level of node n is the number of nodes on the path from root to node n. DefiniFon: The level of node n in a tree T –  If n is the root of T, the level of n is 1. –  If n is not the root of T, its level is 1 greater than the level of its parent. 3/7/11 CS202  ­ Fundamentals of Computer Science II 5 Height of A Tree Height – The number of nodes on the longest path from the root to a leaf. •  The height of a tree T in terms of the levels of its nodes is defined as: –  If T is empty, its height is 0 –  If T is not empty, its height is equal to the maximum level of its nodes. •  Or, the height of a tree T can be defined as recursively as: –  If T is empty, its height is 0. –  If T is non ­empty tree, then since T is of the form: r T1 T2 Tk height(T) = 1 + max{height(T1),height(T2),...,height(Tk)} 3/7/11 CS202  ­ Fundamentals of Computer Science II 6 Binary Tree •  A binary tree T is a set of nodes with the following proper4es: –  The set can be empty. –  Otherwise, the set is par44oned into three disjoint subsets: •  a tree consists of a dis4nguished node r, called root, and •  two possibly empty sets are binary tree, called le< and right subtrees of r. •  T is a binary tree if either –  T has no nodes, or –  T is of the form: r TL TR where r is a node and TL and TR are binary trees. 3/7/11 CS202  ­ Fundamentals of Computer Science II 7 Binary Tree Terminology Le< Child – The lee child of node n is a node directly below and to the lee of node n in a binary tree. Right Child – The right child of node n is a node directly below and to the right of node n in a binary tree. Le< Subtree – In a binary tree, the lee subtree of node n is the lee child (if any) of node n plus its descendants. Right Subtree – In a binary tree, the right subtree of node n is the right child (if any) of node n plus its descendants. 3/7/11 CS202  ­ Fundamentals of Computer Science II 8 Binary Tree  ­ ­ Example •  A is the root. •  B is the left child of A, and C is the right child of A. •  D doesn t have a right child. •  H doesn t have a left child. •  B, F, G and I are leaves. •A •C •B •D •F •E •G •H •I 3/7/11 CS202  ­ Fundamentals of Computer Science II 9 Binary Tree – RepresenEng Algebraic Expressions 3/7/11 CS202  ­ Fundamentals of Computer Science II 10 Height of Binary Tree •  The height of a binary tree T can be defined as recursively as: –  If T is empty, its height is 0. –  If T is non ­empty tree, then since T is of the form r TL TR the height of T is 1 greater than the height of its root s taller subtree; ie. height(T) = 1 + max{height(TL),height(TR)} 3/7/11 CS202  ­ Fundamentals of Computer Science II 11 Height of Binary Tree (cont.) Binary trees with the same nodes but different heights 3/7/11 CS202  ­ Fundamentals of Computer Science II 12 Number of Binary trees with Same # of Nodes empty tree (1 tree) • n=0 n=1 • n=2 • • • • n=3 • • (2 trees) • • • • • • • • • (5 trees) • • • ( n!1)/ 2 n is even NumBT ( N ) = 2 " ( NumBT (i ) NumBT (n ! i ! 1)) i=0 (( n!1)/ 2 )!1 NumBT ( N ) = 2 n is odd " ( NumBT (i ) NumBT (n ! i ! 1)) i=0 + NumBT ((n ! 1) / 2 ) NumBT ((n ! 1) / 2 ) 3/7/11 CS202  ­ Fundamentals of Computer Science II 13 Full Binary Tree •  In a full binary tree of height h, all nodes that are at a level less than h have two children each. •  Each node in a full binary tree has lee and right subtrees of the same height. •  Among binary trees of height h, a full binary tree has as many leaves as possible, and they all are at level h. •  A full binary has no missing nodes. •  Recursive definiFon of full binary tree: –  If T is empty, T is a full binary tree of height 0. –  If T is not empty and has height h>0, T is a full binary tree if its root s subtrees are both full binary trees of height h ­1. 3/7/11 CS202  ­ Fundamentals of Computer Science II 14 Full Binary Tree – Example A full binary tree of height 3 3/7/11 CS202  ­ Fundamentals of Computer Science II 15 Complete Binary Tree •  A complete binary tree of height h is a binary tree that is full down to level h ­1, with level h filled in from lee to right. •  A binary tree T of height h is complete if 1.  All nodes at level h ­2 and above have two children each, and 2.  When a node at level h ­1 has children, all nodes to its lee at the same level have two children each, and 3.  When a node at level h ­1 has one child, it is a lee child. –  A full binary tree is a complete binary tree. 3/7/11 CS202  ­ Fundamentals of Computer Science II 16 Complete Binary Tree – Example 3/7/11 CS202  ­ Fundamentals of Computer Science II 17 Balanced Binary Tree •  A binary tree is height balanced (or balanced), if the height of any node s right subtree differs from the height of the node s lee subtree by no more than 1. •  A complete binary tree is a balanced tree. •  Later, we look at other height balanced trees. –  AVL trees –  Red ­Black trees, .... 3/7/11 CS202  ­ Fundamentals of Computer Science II 18 Maximum and Minimum Heights of a Binary Tree •  The efficiency of most of the binary tree operaFons depends on the height of the tree. •  The maximum number of key comparisons for retrieval, deleFon, and inserFon operaFons for BSTs is the height of the tree. •  The maximum of height of a binary tree with n nodes is n. •  Each level of a minimum height tree, except the last level, must contain as many nodes as possible. 3/7/11 CS202  ­ Fundamentals of Computer Science II 19 Maximum and Minimum Heights of a Binary Tree A maximum-height binary tree with seven nodes 3/7/11 Some binary trees of height 3 CS202  ­ Fundamentals of Computer Science II 20 CounEng the nodes in a full binary tree of height h 3/7/11 CS202  ­ Fundamentals of Computer Science II 21 Some Height Theorems Theorem: A full binary of height h≥0 has 2h ­1 nodes. –  The maximum number of nodes that a binary tree of height h can have is 2h ­1.  We cannot insert a new node into a full binary tree without increasing its height. 3/7/11 CS202  ­ Fundamentals of Computer Science II 22 Some Height Theorems Theorem 10 ­4: The minimum height of a binary tree with n nodes is Ⱥlog2(n+1)Ⱥ . Proof: Let h be the smallest integer such that n≤2h ­1. We can establish following facts: Fact 1 – A binary tree whose height is ≤ h ­1 has < n nodes. –  Otherwise h cannot be smallest integer in our assumpFon. Fact 2 – There exists a complete binary tree of height h that has exactly n nodes. –  A full binary tree of height h ­1 has 2h ­1 ­1 nodes. –  Since a binary tree of height h cannot have more than 2h ­1 nodes. –  At level h, we will reach n nodes. Fact 3 – The minimum height of a binary tree with n nodes is the smallest integer h such that n ≤2h ­1. So, 2h ­1 ­1 < n ≤ 2h ­1 2h ­1 < n+1 ≤ 2h h ­1 < log2(n+1) ≤ h Thus, h = Ⱥlog2(n+1)Ⱥ is the minimum height of a binary tree with n nodes. 3/7/11 CS202  ­ Fundamentals of Computer Science II 23 An Array ­Based ImplementaEon of Binary Trees const int MAX_NODES = 100; !// maximum number of nodes typedef string TreeItemType;! class TreeNode { ! ! !// node in the tree private: !TreeNode(); !TreeNode(const TreeItemType& nodeItem, int left, int right); ! !TreeItemType item; !int leftChild; ! !int rightChild; ! ! ! !// data portion !// index to left child !// index to right child !// friend class - can access private parts !friend class BinaryTree; }; // An array of tree nodes TreeNode[MAX_NODES] tree; int root; int free; 3/7/11 CS202  ­ Fundamentals of Computer Science II 24 An Array ­Based ImplementaEon (cont.) •  A free list keeps track of available nodes. •  To insert a new node into the tree, we first obtain an available node from the free list. •  When we delete a node from the tree, we have to place into the free list so that we can use it later. 3/7/11 CS202  ­ Fundamentals of Computer Science II 25 An Array ­Based RepresentaEon of a Complete Binary Tree •  If we know that our binary tree is a complete binary tree, we can use a simpler array ­based representaFon for complete binary trees •  without using leeChild and rightChild links •  We can number the nodes level by level, and lee to right (starFng from 0, the root will be 0). If a node is numbered as i, in the ith locaFon of the array, tree[i], contains this node without links. •  Using these numbers we can find leeChild, rightChild, and parent of a node i. The lee child (if it exists) of node i is tree[2*i+1]! The right child (if it exists) of node i is tree[2*i+2]! The parent (if it exists) of node i is 3/7/11 CS202  ­ Fundamentals of Computer Science II tree[(i-1)/2]! 26 An Array ­Based RepresentaEon of a Complete Binary Tree (cont.) 0 1 3 3/7/11 2 4 5 CS202  ­ Fundamentals of Computer Science II 27 Pointer ­Based ImplementaEon of Binary Trees 3/7/11 CS202  ­ Fundamentals of Computer Science II 28 A Pointer ­Based ImplementaEon of a Binary Tree Node typedef string TreeItemType;! class TreeNode { // node in the tree! private:! TreeNode() {}! TreeNode(const TreeItemType& nodeItem,! TreeNode *left = NULL,! TreeNode *right = NULL)! :item(nodeItem),leftChildPtr(left),rightChildPtr(right) {}! TreeItemType item; // data portion! TreeNode *leftChildPtr; // pointer to left child! TreeNode *rightChildPtr; // pointer to right child! friend class BinaryTree;! }; ! 3/7/11 CS202  ­ Fundamentals of Computer Science II 29 Binary Tree – TreeExcepEon.h class TreeException : public exception{ private: string msg; public: !virtual const char* what() const throw() !{ ! !return msg.c_str(); !}! TreeException(const string & message =""): !exception(), msg(message) {};! !~TreeException() throw() {}; }; // end TreeException 3/7/11 CS202  ­ Fundamentals of Computer Science II 30 The BinaryTree Class •  ProperFes –  TreeNode * root! •  Constructors –  BinaryTree();! –  BinaryTree(const TreeItemType& rootItem);! –  BinaryTree(const TreeItemType& rootItem, ! !! ! BinaryTree& leftTree, BinaryTree& rightTree);! –  BinaryTree(const BinaryTree& tree);! • void copyTree(TreeNode *treePtr, TreeNode* & newTreePtr) const;! •  Destructor –  ~BinaryTree();! • void destroyTree(TreeNode * &treePtr);! 3/7/11 CS202  ­ Fundamentals of Computer Science II 31 BinaryTree: Public Methods •  •  •  •  •  •  •  •  •  •  •  •  •  •  bool isEmpty()! TreeItemType rootData() const throw(TreeException)! void setRootData(const TreeItemType& newItem)! void attachLeft(const TreeItemType& newItem)! void attachRight(const TreeItemType& newItem)! void attachLeftSubtree(BinaryTree& leftTree)! void attachRightSubtree(BinaryTree& rightTree)! void detachLeftSubtree(BinaryTree& leftTree)! void detachRightSubtree(BinaryTree& rightTree)! BinaryTree leftSubtree()! BinaryTree rightSubtree()! void preorderTraverse(FunctionType visit_fn)! void inorderTraverse(FunctionType visit_fn)! void postorderTraverse(FunctionType visit_fn)! •  FunctionType is a pointer to a funcFon: •  typedef void (*FunctionType)(TreeItemType& anItem);! 3/7/11 CS202  ­ Fundamentals of Computer Science II 32 BinaryTree: ImplementaEon •  The complete implementaFon is in your text book •  In class, we will go through only some methods –  Skipping straighoorward methods •  Such as isEmpty, rootData, and setRootData funcFons –  Skipping some details •  Such as throwing excepFons 3/7/11 CS202  ­ Fundamentals of Computer Science II 33 // Default constructor! BinaryTree::BinaryTree() : root(NULL) {! }! // Protected constructor! BinaryTree::BinaryTree(TreeNode *nodePtr) : root(nodePtr) {! }! // Constructor! BinaryTree::BinaryTree(const TreeItemType& rootItem) {! ! !root = new TreeNode(rootItem, NULL, NULL);! }! 3/7/11 CS202  ­ Fundamentals of Computer Science II 34 // Constructor! BinaryTree::BinaryTree(const TreeItemType& rootItem, ! ! ! ! ! BinaryTree& leftTree, BinaryTree& rightTree) {! ! !root = new TreeNode(rootItem, NULL, NULL);! ! !attachLeftSubtree(leftTree);! ! !attachRightSubtree(rightTree);! }! void BinaryTree::attachLeftSubtree(BinaryTree& leftTree) {! ! !// Assertion: nonempty tree; no left child! ! !if (!isEmpty() && (root->leftChildPtr == NULL)) {! ! ! !root->leftChildPtr = leftTree.root;! ! ! !leftTree.root = NULL! ! !}! }! void BinaryTree::attachRightSubtree(BinaryTree& rightTree) {! ! !// Left as an exercise! }! 3/7/11 CS202  ­ Fundamentals of Computer Science II 35 // Copy constructor! BinaryTree::BinaryTree(const BinaryTree& tree) {! ! !copyTree(tree.root, root);! }! // Uses preorder traversal for the copy operation! // (Visits first the node and then the left and right children)! void BinaryTree::copyTree(TreeNode *treePtr, TreeNode *& newTreePtr) const { ! ! ! ! ! ! ! ! }! 3/7/11 !if (treePtr != NULL) { ! !// copy node! ! !newTreePtr = new TreeNode(treePtr->item, NULL, NULL);! ! !copyTree(treePtr->leftChildPtr, newTreePtr->leftChildPtr);! ! !copyTree(treePtr->rightChildPtr, newTreePtr->rightChildPtr);! !}! !else! ! !newTreePtr = NULL; !// copy empty tree! CS202  ­ Fundamentals of Computer Science II 36 // Destructor! BinaryTree::~BinaryTree() {! ! !destroyTree(root);! }! // Uses postorder traversal for the destroy operation! // (Visits first the left and right children and then the node)! void BinaryTree::destroyTree(TreeNode *& treePtr) {! ! ! ! ! ! ! }! 3/7/11 !if (treePtr != NULL){! ! !destroyTree(treePtr->leftChildPtr);! ! !destroyTree(treePtr->rightChildPtr);! ! !delete treePtr;! ! !treePtr = NULL;! !}! CS202  ­ Fundamentals of Computer Science II 37 Binary Tree Traversals •  Preorder Traversal –  The node is visited before its lee and right subtrees, •  Postorder Traversal –  The node is visited aeer both subtrees. •  Inorder Traversal –  The node is visited between the subtrees, –  Visit lee subtree, visit the node, and visit the right subtree. 3/7/11 CS202  ­ Fundamentals of Computer Science II 38 Binary Tree Traversals 3/7/11 CS202  ­ Fundamentals of Computer Science II 39 void BinaryTree::preorderTraverse(FunctionType visit) {! ! !preorder(root, visit);! }! void BinaryTree::inorderTraverse(FunctionType visit) {! ! !inorder(root, visit);! }! void BinaryTree::postorderTraverse(FunctionType visit) {! ! !postorder(root, visit);! }!  ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ ­ Remember that FunctionType is a pointer to a funcFon •  Variables that point to the address of a funcFon •  typedef void (*FunctionType)(TreeItemType& anItem);! Example of using inorderTraverse funcFon •  void display(TreeItemType& anItem) { cout << anItem << endl; } •  BinaryTree T1;! !T1.inorderTraverse(display);! 3/7/11 CS202  ­ Fundamentals of Computer Science II 40 void BinaryTree::preorder(TreeNode *treePtr, FunctionType visit) {! ! !if (treePtr != NULL) {! ! ! !visit(treePtr->item);! ! ! !preorder(treePtr->leftChildPtr, visit);! ! ! !preorder(treePtr->rightChildPtr, visit);! ! !}! }! void BinaryTree::inorder(TreeNode *treePtr, FunctionType visit) {! ! !if (treePtr != NULL) {! ! ! !inorder(treePtr->left...
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