BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE

Transcription

1 BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE CSI 33 Section E01 Binary Search Trees The Binary Search Tree Property Binary Search Trees, or BSTs, are binary trees with the The Binary Search Tree Property. The binary search tree property states that any value in the left subtree below a node is less than the item value in that node, and that any value in the right subtree below a node is greater. This means that if the value being searched for is not found at that node then one only needs to search one of the subtrees below that node. In the average case, this means one reduces the search space by half for each node actually inspected. As with binary search, this leads to average search time of log(n). BST Binary Search Tree. For any node, values in its left subtree are less than its own item value, while values in its right subtree are greater than its own item value. +_root: TreeNode * NULL value if tree is empty +BST() +insert(value:int): void +find(value:int): int +delete_(value:int): void -_insertrec(t:treenode *, value:int): TreeNode * -_findrec(t:treenode *, value:int): TreeNode * -_deleterec(t:treenode *, value:int): TreeNode * -_deletemax(t:treenode *, item:int&): TreeNode * 1

2 BST.h #pragma once // BST.h #include "TreeNode.h" #define max(x,y) (x > y? x:y) // A BST is a binary search tree. class BST friend std::ostream& operator<<(std::ostream& os, BST& t); public: BST(); BST(); void insert(int value); int find(int value); void delete (int value); private: TreeNode * root; TreeNode * insertrec(treenode* t, int value); TreeNode * findrec(treenode* t, int value); TreeNode * deleterec(treenode* t, int value); TreeNode * deletemax(treenode *pnode, int& item); // item is updated as side-effect void deletenodes(treenode *t); ; 2

3 insert void BST::insert(int value) /* pre: value is not in tree post: value is in tree */ root = insertrec( root, value); TreeNode *BST:: insertrec(treenode* t, int value) if (t == NULL) t = new TreeNode(value, NULL, NULL); else if (value < t-> item) t-> left = insertrec(t-> left, value); else if (value > t-> item) t-> right = insertrec(t-> right, value); else //raise exception since value is already in tree string s = "Can t insert value " + to string(value) + ": already in tree"; throw(std::out of range(s.c str())); //update tree height t-> height = max(getheight(t-> left), getheight(t-> right)) + 1; 3

4 find int BST::find(int value) /* pre: item is in tree post: Returns item (todo: return value associated with key = item) */ TreeNode *t = findrec( root, value); return t-> item; TreeNode *BST:: findrec(treenode *t, int value) if (t == NULL) //raise exception since value not in tree string s = "Can t find item " + to string(value) + ": throw(std::out of range(s.c str())); else if (value < t-> item) return findrec(t-> left, value); else if (value > t-> item) return findrec(t-> right, value); else // found value not in tree"; 4

5 delete void BST::delete (int value) /* pre: item is in tree post: item is not in tree */ root = deleterec( root, value); TreeNode *BST:: deleterec(treenode *t, int value) if (t == NULL) //raise exception since value not in tree string s = "Can t delete item " + to string(value) + ": throw(std::out of range(s.c str())); else if (value < t-> item) // deletion from left t- left = deleterec(t-> left, value); else if (value > t-> item) // deletion from right not in tree"; t-> right = deleterec(t-> right, value); // deletion from right else // delete t if (t-> left == NULL) //promote right subtree TreeNode *new t = t-> right; t = new t; else if (t-> right == NULL) //promote left subtree TreeNode *new t = t-> left; t = new t; else // node t can t be deleted--overwrite it with max item value of left subtree // then delete node where max item value was found t-> left = deletemax(t-> left, t-> item);

6 // update height of tree rooted at t if (t!= NULL) t-> height = max(getheight(t-> left), getheight(t-> right)) + 1; TreeNode *BST:: deletemax(treenode *t, int& item) //use pass-by-ref to overwrite deleted item value if (t-> right == NULL) // t is the max TreeNode *max left = t-> left; item = t-> item; return max left; else // max is in right sub tree--recursively find and delete it t-> right = deletemax(t-> right, item);

7 Deleting a value (8) when neither subtree is NULL else // delete t if (t-> left == NULL) //promote right subtree TreeNode *new t = t-> right; t = new t; else if (t-> right == NULL) //promote left subtree TreeNode *new t = t-> left; t = new t; else // can t delete t-overwrite max t-> left = deletemax(t-> left, t-> item); Before first call to deletemax from deleterec: 2 8 t >_item t >_left

8 TreeNode *BST:: deletemax(treenode *t, int& item) if (t-> right == NULL) // t has the max TreeNode *max left = t-> left; item = t-> item; return max left; else // max is in right subtree t-> right = deletemax(t-> right, item); After first recursive call to deletemax: 8 &item t 4 3 t >_right 8

9 TreeNode *BST:: deletemax(treenode *t, int& item) if (t-> right == NULL) // t has the max TreeNode *max left = t-> left; item = t-> item; return max left; else // max is in right subtree t-> right = deletemax(t-> right, item); After TreeNode *max left = t-> left; and item = t-> item; &item t max_left 9

10 After return max left; &item? t max_left 10

11 After t-> right = deletemax(t-> right, item); &item t 4 3 t >_right 11

12 else // delete t if (t-> left == NULL) //promote right subtree TreeNode *new t = t-> right; t = new t; else if (t-> right == NULL) //promote left subtree TreeNode *new t = t-> left; t = new t; else // can t delete t-overwrite max t-> left = After t-> left = 2 deletemax(t-> left, t-> item); deletemax(t-> left, t-> item); t t >_left