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Trees, Binary Trees, and Binary Search Trees

COMP171. Trees, Binary Trees, and Binary Search Trees. Trees. Linear access time of linked lists is prohibitive Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)? Trees Basic concepts Tree traversal Binary tree

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Trees, Binary Trees, and Binary Search Trees

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  1. COMP171 Trees, Binary Trees, and Binary Search Trees

  2. Trees • Linear access time of linked lists is prohibitive • Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)? • Trees • Basic concepts • Tree traversal • Binary tree • Binary search tree and its operations

  3. Trees • A tree T is a collection of nodes • T can be empty • (recursive definition) If not empty, a tree T consists of • a (distinguished) node r (the root), • and zero or more nonempty subtrees T1, T2, ...., Tk

  4. Tree can be viewed as a ‘nested’ lists • Tree is also a graph …

  5. Some Terminologies • Child and Parent • Every node except the root has one parent  • A node can have an zero or more children • Leaves • Leaves are nodes with no children • Sibling • nodes with same parent

  6. More Terminologies • Path • A sequence of edges • Length of a path • number of edges on the path • Depth of a node • length of the unique path from the root to that node • Height of a node • length of the longest path from that node to a leaf • all leaves are at height 0 • The height of a tree = the height of the root = the depth of the deepest leaf • Ancestor and descendant • If there is a path from n1 to n2 • n1 is an ancestor of n2, n2 is a descendant of n1 • Proper ancestor and proper descendant

  7. Example: UNIX Directory

  8. Example: Expression Trees • Leaves are operands (constants or variables) • The internal nodes contain operators • Will not be a binary tree if some operators are not binary

  9. Tree Traversal • Used to print out the data in a tree in a certain order • Pre-order traversal • Print the data at the root • Recursively print out all data in the leftmost subtree • … • Recursively print out all data in the rightmost subtree

  10. Preorder, Postorder and Inorder • Preorder traversal • node, left, right • prefix expression • ++a*bc*+*defg

  11. Postorder traversal left, right, node postfix expression abc*+de*f+g*+ Inorder traversal left, node, right infix expression a+b*c+d*e+f*g Preorder, Postorder and Inorder

  12. Example: Unix Directory Traversal PreOrder PostOrder

  13. Preorder, Postorder and Inorder Pseudo Code

  14. Binary Trees • A tree in which no node can have more than two children • The depth of an “average” binary tree is considerably smaller than N, even though in the worst case, the depth can be as large as N – 1. Generic binary tree Worst-casebinary tree

  15. Convert a Generic Tree to a Binary Tree

  16. A Binary Tree Node template <class Comparable> class BinaryTreeNode { Comparable element; // the data BinaryTreeNode* left; // left child BinaryTreeNode* right; // right child public: BinaryTreeNode getleft () { return left; } BinaryTreeNode getright () { return right; } }; template<class Comparable> void preorder (BinaryTreeNode<Comparable> *root) { if (!root){ // output the node preorder (root->getleft()); preorder (root->getright()); } }

  17. Binary Search Trees (BST) • A data structure for efficient searching, inser-tion and deletion • Binary search tree property • For every node X • All the keys in its left subtree are smaller than the key value in X • All the keys in its right subtree are larger than the key value in X

  18. Binary Search Trees A binary search tree Not a binary search tree

  19. Binary Search Trees The same set of keys may have different BSTs • Average depth of a node is O(log N) • Maximum depth of a node is O(N)

  20. Searching BST • If we are searching for 15, then we are done. • If we are searching for a key < 15, then we should search in the left subtree. • If we are searching for a key > 15, then we should search in the right subtree.

  21. Searching (Find) • Find X: return a pointer to the node that has key X, or NULL if there is no such node • Time complexity: O(height of the tree)

  22. Inorder Traversal of BST • Inorder traversal of BST prints out all the keys in sorted order Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

  23. findMin/ findMax • Goal: return the node containing the smallest (largest) key in the tree • Algorithm: Start at the root and go left (right) as long as there is a left (right) child. The stopping point is the smallest (largest) element • Time complexity = O(height of the tree)

  24. Insertion • Proceed down the tree as you would with a find • If X is found, do nothing (or update something) • Otherwise, insert X at the last spot on the path traversed • Time complexity = O(height of the tree)

  25. void insert(double x, BinaryNode*& t) { if (t==NULL) t = new BinaryNode(x,NULL,NULL); else if (x<t->element) insert(x,t->left); else if (t->element<x) insert(x,t->right); else ; // do nothing }

  26. Deletion • When we delete a node, we need to consider how we take care of the children of the deleted node. • This has to be done such that the property of the search tree is maintained.

  27. Deletion under Different Cases • Case 1: the node is a leaf • Delete it immediately • Case 2: the node has one child • Adjust a pointer from the parent to bypass that node

  28. Deletion Case 3 • Case 3: the node has 2 children • Replace the key of the node with the minimum element at the right subtree • Delete that node with minimum element • Has either no child or only right child because if otherwise, that node would not have been the minimum in the first place! So, invoke case 1 or 2. • Time complexity = O(height of the tree)

  29. void remove(double x, BinaryNode*& t) { if (t==NULL) return; // empty tree or not found if (x<t->element) remove(x,t->left); else if (t->element < x) remove (x, t->right); else if (t->left != NULL && t->right != NULL) // two children { t->element = finMin(t->right) ->element; remove(t->element,t->right); } else { Binarynode* oldNode = t; t = (t->left != NULL) ? t->left : t->right; delete oldNode; } }

  30. Make a binary or BST ADT …

  31. For a generic (binary) tree: Struct Node { double element; // the data Node* left; // left child Node* right; // right child } class Tree { public: Tree(); // constructor Tree(const Tree& t); ~Tree(); // destructor bool empty() const; double root(); // decomposition (access functions) Tree& left(); Tree& right(); void insert(const double x); // compose x into a tree void remove(const double x); // decompose x from a tree private: Node* root; } access, selection update (insert and remove are different from those of BST)

  32. For BST tree (non-template version) Struct Node { double element; // the data Node* left; // left child Node* right; // right child } class BST { public: BST(); // constructor BST(const Tree& t); ~BST(); // destructor bool empty() const; double root(); // decomposition (access functions) BST left(); BST right(); bool serch(const double x); // search an element void insert(const double x); // compose x into a tree void remove(const double x); // decompose x from a tree private: Node* root; } access, selection update BST is for efficient search, insertion and removal, so restricting these functions.

  33. Weiss textbook: class BST { public: BST(); BST(const Tree& t); ~BST(); bool empty() const; bool search(const double x); // contains void insert(const double x); // compose x into a tree void remove(const double x); // decompose x from a tree private: Struct Node { double element; Node* left; Node* right; Node(…) {…}; // constructuro for Node } Node* root; void insert(const double x, Node*& t) const; // recursive function void remove(…) Node* findMin(Node* t); void makeEmpty(Node*& t); // recursive ‘destructor’ bool contains(const double x, Node* t) const; }

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