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如何在Binary Search Tree中实现减小键或更改键?

本文概述

给定二叉搜索树, 编写一个函数, 该函数以以下三个作为参数:

1)树的根

2)旧键值

3)新的关键值

该功能应将旧键值更改为新键值。该函数可以假定二叉搜索树中始终存在旧键值。

例子:

Input: Root of below tree
              50
           /    \
          30      70
         / \    / \
       20   40  60   80 
     Old key value:  40
     New key value:  10

Output: BST should be modified to following
              50
           /    \
          30      70
         /      / \
       20      60   80  
       /
     10

我们强烈建议你最小化浏览器, 然后先尝试一下

这个想法是为旧键值调用delete, 然后为新键值调用insert。以下是该想法的C ++实现。

C ++

//C++ program to demonstrate decrease
//key operation on binary search tree 
#include<bits/stdc++.h> 
  
using namespace std;
  
class node 
{ 
     public :
     int key; 
     node *left, *right; 
}; 
  
//A utility function to 
//create a new BST node 
node *newNode( int item) 
{ 
     node *temp = new node; 
     temp->key = item; 
     temp->left = temp->right = NULL; 
     return temp; 
} 
  
//A utility function to 
//do inorder traversal of BST 
void inorder(node *root) 
{ 
     if (root != NULL) 
     { 
         inorder(root->left); 
         cout <<root->key <<" " ; 
         inorder(root->right); 
     } 
} 
  
/* A utility function to insert 
a new node with given key in BST */
node* insert(node* node, int key) 
{ 
     /* If the tree is empty, return a new node */
     if (node == NULL) return newNode(key); 
  
     /* Otherwise, recur down the tree */
     if (key <node->key) 
         node->left = insert(node->left, key); 
     else
         node->right = insert(node->right, key); 
  
     /* return the (unchanged) node pointer */
     return node; 
} 
  
/* Given a non-empty binary search tree, return the node with minimum key value
found in that tree. Note that the entire
tree does not need to be searched. */
node * minValueNode(node* Node) 
{ 
     node* current = Node; 
  
     /* loop down to find the leftmost leaf */
     while (current->left != NULL) 
         current = current->left; 
  
     return current; 
} 
  
/* Given a binary search tree and 
a key, this function deletes the key 
and returns the new root */
node* deleteNode(node* root, int key) 
{ 
     //base case 
     if (root == NULL) return root; 
  
     //If the key to be deleted is
     //smaller than the root's key, //then it lies in left subtree 
     if (key <root->key) 
         root->left = deleteNode(root->left, key); 
  
     //If the key to be deleted is 
     //greater than the root's key, //then it lies in right subtree 
     else if (key> root->key) 
         root->right = deleteNode(root->right, key); 
  
     //if key is same as root's 
     //key, then This is the node 
     //to be deleted 
     else
     { 
         //node with only one child or no child 
         if (root->left == NULL) 
         { 
             node *temp = root->right; 
             free (root); 
             return temp; 
         } 
         else if (root->right == NULL) 
         { 
             node *temp = root->left; 
             free (root); 
             return temp; 
         } 
  
         //node with two children: Get 
         //the inorder successor (smallest 
         //in the right subtree) 
         node* temp = minValueNode(root->right); 
  
         //Copy the inorder successor's 
         //content to this node 
         root->key = temp->key; 
  
         //Delete the inorder successor 
         root->right = deleteNode(root->right, temp->key); 
     } 
     return root; 
} 
  
//Function to decrease a key
//value in Binary Search Tree 
node *changeKey(node *root, int oldVal, int newVal) 
{ 
     //First delete old key value 
     root = deleteNode(root, oldVal); 
  
     //Then insert new key value 
     root = insert(root, newVal); 
  
     //Return new root 
     return root; 
} 
  
//Driver code 
int main() 
{ 
     /* Let us create following BST 
             50 
         /\ 
         30 70 
         /\ /\ 
     20 40 60 80 */
     node *root = NULL; 
     root = insert(root, 50); 
     root = insert(root, 30); 
     root = insert(root, 20); 
     root = insert(root, 40); 
     root = insert(root, 70); 
     root = insert(root, 60); 
     root = insert(root, 80); 
  
  
     cout <<"Inorder traversal of the given tree \n" ; 
     inorder(root); 
  
     root = changeKey(root, 40, 10); 
  
     /* BST is modified to 
             50 
         /\ 
         30 70 
         //\ 
     20 60 80 
     /
     10 */
     cout <<"\nInorder traversal of the modified tree \n" ; 
     inorder(root); 
  
     return 0; 
} 
  
//This code is contributed by rathbhupendra

C

//C program to demonstrate decrease  key operation on binary search tree
#include<stdio.h>
#include<stdlib.h>
  
struct node
{
     int key;
     struct node *left, *right;
};
  
//A utility function to create a new BST node
struct node *newNode( int item)
{
     struct node *temp =  ( struct node *) malloc ( sizeof ( struct node));
     temp->key = item;
     temp->left = temp->right = NULL;
     return temp;
}
  
//A utility function to do inorder traversal of BST
void inorder( struct node *root)
{
     if (root != NULL)
     {
         inorder(root->left);
         printf ( "%d " , root->key);
         inorder(root->right);
     }
}
  
/* A utility function to insert a new node with given key in BST */
struct node* insert( struct node* node, int key)
{
     /* If the tree is empty, return a new node */
     if (node == NULL) return newNode(key);
  
     /* Otherwise, recur down the tree */
     if (key <node->key)
         node->left  = insert(node->left, key);
     else
         node->right = insert(node->right, key);
  
     /* return the (unchanged) node pointer */
     return node;
}
  
/* Given a non-empty binary search tree, return the node with minimum
    key value found in that tree. Note that the entire tree does not
    need to be searched. */
struct node * minValueNode( struct node* node)
{
     struct node* current = node;
  
     /* loop down to find the leftmost leaf */
     while (current->left != NULL)
         current = current->left;
  
     return current;
}
  
/* Given a binary search tree and a key, this function deletes the key
    and returns the new root */
struct node* deleteNode( struct node* root, int key)
{
     //base case
     if (root == NULL) return root;
  
     //If the key to be deleted is smaller than the root's key, //then it lies in left subtree
     if (key <root->key)
         root->left = deleteNode(root->left, key);
  
     //If the key to be deleted is greater than the root's key, //then it lies in right subtree
     else if (key> root->key)
         root->right = deleteNode(root->right, key);
  
     //if key is same as root's key, then This is the node
     //to be deleted
     else
     {
         //node with only one child or no child
         if (root->left == NULL)
         {
             struct node *temp = root->right;
             free (root);
             return temp;
         }
         else if (root->right == NULL)
         {
             struct node *temp = root->left;
             free (root);
             return temp;
         }
  
         //node with two children: Get the inorder successor (smallest
         //in the right subtree)
         struct node* temp = minValueNode(root->right);
  
         //Copy the inorder successor's content to this node
         root->key = temp->key;
  
         //Delete the inorder successor
         root->right = deleteNode(root->right, temp->key);
     }
     return root;
}
  
//Function to decrease a key value in Binary Search Tree
struct node *changeKey( struct node *root, int oldVal, int newVal)
{
     // First delete old key value
     root = deleteNode(root, oldVal);
  
     //Then insert new key value
     root = insert(root, newVal);
  
     //Return new root
     return root;
}
  
//Driver Program to test above functions
int main()
{
     /* Let us create following BST
               50
            /    \
           30      70
          / \    / \
        20   40  60   80 */
     struct node *root = NULL;
     root = insert(root, 50);
     root = insert(root, 30);
     root = insert(root, 20);
     root = insert(root, 40);
     root = insert(root, 70);
     root = insert(root, 60);
     root = insert(root, 80);
  
  
     printf ( "Inorder traversal of the given tree \n" );
     inorder(root);
  
     root = changeKey(root, 40, 10);
  
     /* BST is modified to
               50
            /    \
           30      70
          /      / \
        20      60   80
        /
      10     */
     printf ( "\nInorder traversal of the modified tree \n" );
     inorder(root);
  
     return 0;
}

Java

//Java program to demonstrate decrease 
//key operation on binary search tree 
class GfG 
{
  
static class node 
{ 
     int key; 
     node left, right; 
}
static node root = null ;
  
//A utility function to 
//create a new BST node 
static node newNode( int item) 
{ 
     node temp = new node(); 
     temp.key = item; 
     temp.left = null ;
     temp.right = null ; 
     return temp; 
} 
  
//A utility function to 
//do inorder traversal of BST 
static void inorder(node root) 
{ 
     if (root != null ) 
     { 
         inorder(root.left); 
         System.out.print(root.key + " " ); 
         inorder(root.right); 
     } 
} 
  
/* A utility function to insert 
a new node with given key in BST */
static node insert(node node, int key) 
{ 
     /* If the tree is empty, return a new node */
     if (node == null ) return newNode(key); 
  
     /* Otherwise, recur down the tree */
     if (key <node.key) 
         node.left = insert(node.left, key); 
     else
         node.right = insert(node.right, key); 
  
     /* return the (unchanged) node pointer */
     return node; 
} 
  
/* Given a non-empty binary search tree, return the node with minimum key value 
found in that tree. Note that the entire 
tree does not need to be searched. */
static node minValueNode(node Node) 
{ 
     node current = Node; 
  
     /* loop down to find the leftmost leaf */
     while (current.left != null ) 
         current = current.left; 
  
     return current; 
} 
  
/* Given a binary search tree and 
a key, this function deletes the key 
and returns the new root */
static node deleteNode(node root, int key) 
{ 
     //base case 
     if (root == null ) return root; 
  
     //If the key to be deleted is 
     //smaller than the root's key, //then it lies in left subtree 
     if (key <root.key) 
         root.left = deleteNode(root.left, key); 
  
     //If the key to be deleted is 
     //greater than the root's key, //then it lies in right subtree 
     else if (key> root.key) 
         root.right = deleteNode(root.right, key); 
  
     //if key is same as root's 
     //key, then This is the node 
     //to be deleted 
     else
     { 
         //node with only one child or no child 
         if (root.left == null ) 
         { 
             node temp = root.right; 
             return temp; 
         } 
         else if (root.right == null ) 
         { 
             node temp = root.left; 
             return temp; 
         } 
  
         //node with two children: Get 
         //the inorder successor (smallest 
         //in the right subtree) 
         node temp = minValueNode(root.right); 
  
         //Copy the inorder successor's 
         //content to this node 
         root.key = temp.key; 
  
         //Delete the inorder successor 
         root.right = deleteNode(root.right, temp.key); 
     } 
     return root; 
} 
  
//Function to decrease a key 
//value in Binary Search Tree 
static node changeKey(node root, int oldVal, int newVal) 
{ 
     //First delete old key value 
     root = deleteNode(root, oldVal); 
  
     //Then insert new key value 
     root = insert(root, newVal); 
  
     //Return new root 
     return root; 
} 
  
//Driver code 
public static void main(String[] args) 
{ 
     /* Let us create following BST 
             50 
         /\ 
         30 70 
         /\ /\ 
     20 40 60 80 */
     root = insert(root, 50 ); 
     root = insert(root, 30 ); 
     root = insert(root, 20 ); 
     root = insert(root, 40 ); 
     root = insert(root, 70 ); 
     root = insert(root, 60 ); 
     root = insert(root, 80 ); 
  
  
     System.out.println( "Inorder traversal of the given tree" ); 
     inorder(root); 
  
     root = changeKey(root, 40 , 10 ); 
  
     /* BST is modified to 
             50 
         /\ 
         30 70 
         //\ 
     20 60 80 
     /
     10 */
     System.out.println( "\nInorder traversal of the modified tree " ); 
     inorder(root); 
}
}
  
//This code is contributed by Prerna saini

Python3

# Python3 program to demonstrate decrease key 
# operation on binary search tree 
  
# A utility function to create a new BST node 
class newNode:
      
     def __init__( self , key): 
         self .key = key
         self .left = self .right = None
  
# A utility function to do inorder
# traversal of BST 
def inorder(root):
     if root ! = None :
         inorder(root.left) 
         print (root.key, end = " " ) 
         inorder(root.right)
  
# A utility function to insert a new
# node with given key in BST 
def insert(node, key):
      
     # If the tree is empty, return a new node 
     if node = = None :
         return newNode(key)
  
     # Otherwise, recur down the tree 
     if key <node.key: 
         node.left = insert(node.left, key) 
     else :
         node.right = insert(node.right, key)
  
     # return the (unchanged) node pointer 
     return node
  
# Given a non-empty binary search tree, return 
# the node with minimum key value found in that 
# tree. Note that the entire tree does not 
# need to be searched. 
def minValueNode(node):
     current = node
  
     # loop down to find the leftmost leaf 
     while current.left ! = None : 
         current = current.left
     return current
  
# Given a binary search tree and a key, this 
# function deletes the key and returns the new root 
def deleteNode(root, key):
      
     # base case 
     if root = = None : 
         return root
  
     # If the key to be deleted is smaller than 
     # the root's key, then it lies in left subtree 
     if key <root.key: 
         root.left = deleteNode(root.left, key) 
  
     # If the key to be deleted is greater than 
     # the root's key, then it lies in right subtree 
     elif key> root.key: 
         root.right = deleteNode(root.right, key)
          
     # if key is same as root's key, then 
     # this is the node to be deleted 
     else :
          
         # node with only one child or no child 
         if root.left = = None :
             temp = root.right
             return temp
         elif root.right = = None :
             temp = root.left
             return temp
  
         # node with two children: Get the inorder 
         # successor (smallest in the right subtree) 
         temp = minValueNode(root.right) 
  
         # Copy the inorder successor's content
         # to this node 
         root.key = temp.key 
  
         # Delete the inorder successor 
         root.right = deleteNode(root.right, temp.key)
     return root
  
# Function to decrease a key value in 
# Binary Search Tree 
def changeKey(root, oldVal, newVal):
      
     # First delete old key value 
     root = deleteNode(root, oldVal) 
  
     # Then insert new key value 
     root = insert(root, newVal)
  
     # Return new root 
     return root
  
# Driver Code
if __name__ = = '__main__' :
      
     # Let us create following BST 
     #         50 
     #     /    \ 
     #     30     70 
     #     /\ /\ 
     # 20 40 60 80 
     root = None
     root = insert(root, 50 ) 
     root = insert(root, 30 ) 
     root = insert(root, 20 ) 
     root = insert(root, 40 ) 
     root = insert(root, 70 ) 
     root = insert(root, 60 ) 
     root = insert(root, 80 ) 
  
     print ( "Inorder traversal of the given tree" )
     inorder(root)
  
     root = changeKey(root, 40 , 10 ) 
     print ()
      
     # BST is modified to 
     #         50 
     #     /    \ 
     #     30     70 
     #     /    /\ 
     # 20     60 80 
     # /
     # 10     
     print ( "Inorder traversal of the modified tree" ) 
     inorder(root)
      
# This code is contributed by PranchalK

C#

//C# program to demonstrate decrease 
//key operation on binary search tree 
using System;
  
class GFG 
{
public class node 
{ 
     public int key; 
     public node left, right; 
}
static node root = null ;
  
//A utility function to 
//create a new BST node 
static node newNode( int item) 
{ 
     node temp = new node(); 
     temp.key = item; 
     temp.left = null ;
     temp.right = null ; 
     return temp; 
} 
  
//A utility function to 
//do inorder traversal of BST 
static void inorder(node root) 
{ 
     if (root != null ) 
     { 
         inorder(root.left); 
         Console.Write(root.key + " " ); 
         inorder(root.right); 
     } 
} 
  
/* A utility function to insert 
a new node with given key in BST */
static node insert(node node, int key) 
{ 
     /* If the tree is empty, return a new node */
     if (node == null ) return newNode(key); 
  
     /* Otherwise, recur down the tree */
     if (key <node.key) 
         node.left = insert(node.left, key); 
     else
         node.right = insert(node.right, key); 
  
     /* return the (unchanged) node pointer */
     return node; 
} 
  
/* Given a non-empty binary search tree, return the node with minimum key value 
found in that tree. Note that the entire 
tree does not need to be searched. */
static node minValueNode(node Node) 
{ 
     node current = Node; 
  
     /* loop down to find the leftmost leaf */
     while (current.left != null ) 
         current = current.left; 
  
     return current; 
} 
  
/* Given a binary search tree and 
a key, this function deletes the key 
and returns the new root */
static node deleteNode(node root, int key) 
{ 
     node temp = null ;
      
     //base case 
     if (root == null ) return root; 
  
     //If the key to be deleted is 
     //smaller than the root's key, //then it lies in left subtree 
     if (key <root.key) 
         root.left = deleteNode(root.left, key); 
  
     //If the key to be deleted is 
     //greater than the root's key, //then it lies in right subtree 
     else if (key> root.key) 
         root.right = deleteNode(root.right, key); 
  
     //if key is same as root's 
     //key, then This is the node 
     //to be deleted 
     else
     { 
          
         //node with only one child or no child 
         if (root.left == null ) 
         { 
             temp = root.right; 
             return temp; 
         } 
         else if (root.right == null ) 
         { 
             temp = root.left; 
             return temp; 
         } 
  
         //node with two children: Get 
         //the inorder successor (smallest 
         //in the right subtree) 
         temp = minValueNode(root.right); 
  
         //Copy the inorder successor's 
         //content to this node 
         root.key = temp.key; 
  
         //Delete the inorder successor 
         root.right = deleteNode(root.right, temp.key); 
     } 
     return root; 
} 
  
//Function to decrease a key 
//value in Binary Search Tree 
static node changeKey(node root, int oldVal, int newVal) 
{ 
     //First delete old key value 
     root = deleteNode(root, oldVal); 
  
     //Then insert new key value 
     root = insert(root, newVal); 
  
     //Return new root 
     return root; 
} 
  
//Driver code 
public static void Main(String[] args) 
{ 
     /* Let us create following BST 
             50 
         /\ 
         30 70 
         /\ /\ 
     20 40 60 80 */
     root = insert(root, 50); 
     root = insert(root, 30); 
     root = insert(root, 20); 
     root = insert(root, 40); 
     root = insert(root, 70); 
     root = insert(root, 60); 
     root = insert(root, 80); 
      
     Console.WriteLine( "Inorder traversal " + 
                       "of the given tree " ); 
     inorder(root); 
  
     root = changeKey(root, 40, 10); 
  
     /* BST is modified to 
             50 
         /\ 
         30 70 
         //\ 
     20 60 80 
     /
     10 */
     Console.WriteLine( "\nInorder traversal " + 
                       "of the modified tree" ); 
     inorder(root); 
}
}
  
//This code is contributed by 29AjayKumar

输出如下:

Inorder traversal of the given tree 
20 30 40 50 60 70 80 
Inorder traversal of the modified tree 
10 20 30 50 60 70 80

上述changeKey()的时间复杂度为O(h), 其中h是BST的高度。

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