本文概述

• 建议：在继续解决方案之前, 请先在"实践"上解决它。
• C ++
• C
• Java
• python
• C#

我们已经讨论了BST搜索和插入操作。在这篇文章中，我们讨论了删除操作。当我们删除一个节点时，有三种可能。
【算法设计（二叉搜索树如何实现删除操作（））】1)
要删除的节点是叶子：只需从树上移开即可。

5050 /\delete(20)/\ 3070---------> 3070 /\/\\/\ 20406080406080

2)要删除的节点只有一个孩子：将子节点复制到节点并删除该子节点

5050 /\delete(30)/\ 3070---------> 4070 \/\/\ 4060806080

3)要删除的节点有两个子节点： 查找节点的有序后继者。将有序后继者的内容复制到节点并删除有序后继者。注意, 也可以使用有序的前身。

5060 /\delete(50)/\ 4070---------> 4070 /\\ 608080

要注意的重要一点是, 仅在正确的子代不为空时才需要有序继承者。在这种特定情况下, 可以通过在节点的右子节点中找到最小值来获得有序后继。
推荐：请在"实践首先, 在继续解决方案之前。 C ++

// C++ program to demonstrate delete operation in binary search tree #include< bits/stdc++.h> using namespace std; 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); cout< < 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 & & 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; }// Driver Program to test above functions int main() { /* Let us create following BST 50 /\ 3070 / \ / \ 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); cout < < "Inorder traversal of the given tree \n" ; inorder(root); cout< < "\nDelete 20\n" ; root = deleteNode(root, 20); cout< < "Inorder traversal of the modified tree \n" ; inorder(root); cout< < "\nDelete 30\n" ; root = deleteNode(root, 30); cout< < "Inorder traversal of the modified tree \n" ; inorder(root); cout< < "\nDelete 50\n" ; root = deleteNode(root, 50); cout< < "Inorder traversal of the modified tree \n" ; inorder(root); return 0; }// This code is contributed by shivanisinghss2110

C

// C program to demonstrate delete operation in 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 & & 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; }// Driver Program to test above functions int main() { /* Let us create following BST 50 /\ 3070 /\/\ 20406080 */ 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); printf ( "\nDelete 20\n" ); root = deleteNode(root, 20); printf ( "Inorder traversal of the modified tree \n" ); inorder(root); printf ( "\nDelete 30\n" ); root = deleteNode(root, 30); printf ( "Inorder traversal of the modified tree \n" ); inorder(root); printf ( "\nDelete 50\n" ); root = deleteNode(root, 50); printf ( "Inorder traversal of the modified tree \n" ); inorder(root); return 0; }

Java

// Java program to demonstrate delete operation in binary search tree class BinarySearchTree { /* Class containing left and right child of current node and key value*/ class Node { int key; Node left, right; public Node( int item) { key = item; left = right = null ; } }// Root of BST Node root; // Constructor BinarySearchTree() { root = null ; }// This method mainly calls deleteRec() void deleteKey( int key) { root = deleteRec(root, key); }/* A recursive function to insert a new key in BST */ Node deleteRec(Node root, int key) { /* Base Case: If the tree is empty */ if (root == null )return root; /* Otherwise, recur down the tree */ if (key < root.key) root.left = deleteRec(root.left, key); else if (key > root.key) root.right = deleteRec(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 ) return root.right; else if (root.right == null ) return root.left; // node with two children: Get the inorder successor (smallest // in the right subtree) root.key = minValue(root.right); // Delete the inorder successor root.right = deleteRec(root.right, root.key); }return root; }int minValue(Node root) { int minv = root.key; while (root.left != null ) { minv = root.left.key; root = root.left; } return minv; }// This method mainly calls insertRec() void insert( int key) { root = insertRec(root, key); }/* A recursive function to insert a new key in BST */ Node insertRec(Node root, int key) {/* If the tree is empty, return a new node */ if (root == null ) { root = new Node(key); return root; }/* Otherwise, recur down the tree */ if (key < root.key) root.left = insertRec(root.left, key); else if (key > root.key) root.right = insertRec(root.right, key); /* return the (unchanged) node pointer */ return root; }// This method mainly calls InorderRec() void inorder() { inorderRec(root); }// A utility function to do inorder traversal of BST void inorderRec(Node root) { if (root != null ) { inorderRec(root.left); System.out.print(root.key + " " ); inorderRec(root.right); } }// Driver Program to test above functions public static void main(String[] args) { BinarySearchTree tree = new BinarySearchTree(); /* Let us create following BST 50 /\ 3070 /\/\ 20406080 */ tree.insert( 50 ); tree.insert( 30 ); tree.insert( 20 ); tree.insert( 40 ); tree.insert( 70 ); tree.insert( 60 ); tree.insert( 80 ); System.out.println( "Inorder traversal of the given tree" ); tree.inorder(); System.out.println( "\nDelete 20" ); tree.deleteKey( 20 ); System.out.println( "Inorder traversal of the modified tree" ); tree.inorder(); System.out.println( "\nDelete 30" ); tree.deleteKey( 30 ); System.out.println( "Inorder traversal of the modified tree" ); tree.inorder(); System.out.println( "\nDelete 50" ); tree.deleteKey( 50 ); System.out.println( "Inorder traversal of the modified tree" ); tree.inorder(); } }

python

# Python program to demonstrate delete operation # in binary search tree# A Binary Tree Node class Node:# Constructor to create a new node def __init__( self , key): self .key = key self .left = None self .right = None# A utility function to do inorder traversal of BST def inorder(root): if root is not None : inorder(root.left) print root.key, 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 is None : return Node(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 minum 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 is not None ): current = current.left return current # Given a binary search tree and a key, this function # delete the key and returns the new root def deleteNode(root, key):# Base Case if root is None : return root # If the key to be deleted is smaller than the root's # key then it lies inleft subtree if key < root.key: root.left = deleteNode(root.left, key)# If the kye to be delete 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 is None : temp = root.right root = None return temp elif root.right is None : temp = root.left root = None 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 # Driver program to test above functions """ Let us create following BST 50 /\ 3070 /\/\ 20406080 """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)print "\nDelete 20" root = deleteNode(root, 20 ) print "Inorder traversal of the modified tree" inorder(root)print "\nDelete 30" root = deleteNode(root, 30 ) print "Inorder traversal of the modified tree" inorder(root)print "\nDelete 50" root = deleteNode(root, 50 ) print "Inorder traversal of the modified tree" inorder(root)# This code is contributed by Nikhil Kumar Singh(nickzuck_007)

C#

// C# program to demonstrate delete // operation in binary search tree using System; public class BinarySearchTree { /* Class containing left and right child of current node and key value*/ class Node { public int key; public Node left, right; public Node( int item) { key = item; left = right = null ; } } // Root of BST Node root; // Constructor BinarySearchTree() { root = null ; } // This method mainly calls deleteRec() void deleteKey( int key) { root = deleteRec(root, key); } /* A recursive function to insert a new key in BST */ Node deleteRec(Node root, int key) { /* Base Case: If the tree is empty */ if (root == null ) return root; /* Otherwise, recur down the tree */ if (key < root.key) root.left = deleteRec(root.left, key); else if (key > root.key) root.right = deleteRec(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 ) return root.right; else if (root.right == null ) return root.left; // node with two children: Get the // inorder successor (smallest // in the right subtree) root.key = minValue(root.right); // Delete the inorder successor root.right = deleteRec(root.right, root.key); } return root; } int minValue(Node root) { int minv = root.key; while (root.left != null ) { minv = root.left.key; root = root.left; } return minv; } // This method mainly calls insertRec() void insert( int key) { root = insertRec(root, key); } /* A recursive function to insert a new key in BST */ Node insertRec(Node root, int key) { /* If the tree is empty, return a new node */ if (root == null ) { root = new Node(key); return root; } /* Otherwise, recur down the tree */ if (key < root.key) root.left = insertRec(root.left, key); else if (key > root.key) root.right = insertRec(root.right, key); /* return the (unchanged) node pointer */ return root; } // This method mainly calls InorderRec() void inorder() { inorderRec(root); } // A utility function to do inorder traversal of BST void inorderRec(Node root) { if (root != null ) { inorderRec(root.left); Console.Write(root.key + " " ); inorderRec(root.right); } } // Driver code public static void Main(String[] args) { BinarySearchTree tree = new BinarySearchTree(); /* Let us create following BST 50 / \ 30 70 / \ / \ 20 40 60 80 */ tree.insert(50); tree.insert(30); tree.insert(20); tree.insert(40); tree.insert(70); tree.insert(60); tree.insert(80); Console.WriteLine( "Inorder traversal of the given tree" ); tree.inorder(); Console.WriteLine( "\nDelete 20" ); tree.deleteKey(20); Console.WriteLine( "Inorder traversal of the modified tree" ); tree.inorder(); Console.WriteLine( "\nDelete 30" ); tree.deleteKey(30); Console.WriteLine( "Inorder traversal of the modified tree" ); tree.inorder(); Console.WriteLine( "\nDelete 50" ); tree.deleteKey(50); Console.WriteLine( "Inorder traversal of the modified tree" ); tree.inorder(); } } // This code has been contributed // by PrinciRaj1992

输出如下：

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

插图：

文章图片

文章图片
时间复杂度：删除操作的最坏情况时间复杂度是O(h), 其中h是二叉搜索树的高度。在最坏的情况下, 我们可能必须从根移动到最深的叶节点。倾斜的树的高度可能变为n, 删除操作的时间复杂度可能变为O(n)
针对上述两个子案例的代码的优化：
在上面的递归代码中, 我们递归调用delete()作为后继者。我们可以通过跟踪后继者的父节点来避免递归调用, 这样我们就可以简单地通过将父代的子代设为NULL来删除后继者。我们知道后继者将永远是叶子节点。

// C++ program to implement optimized delete in BST. #include < bits/stdc++.h> using namespace std; struct Node { int key; struct 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); printf ( "%d " , 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 binary search tree and a key, this function deletes the key and returns the new root */ Node* deleteNode(Node* root, int k) { // Base case if (root == NULL) return root; // Recursive calls for ancestors of // node to be deleted if (root-> key > k) { root-> left = deleteNode(root-> left, k); return root; } else if (root-> key < k) { root-> right = deleteNode(root-> right, k); return root; }// We reach here when root is the node // to be deleted.// If one of the children is empty if (root-> left == NULL) { Node* temp = root-> right; delete root; return temp; } else if (root-> right == NULL) { Node* temp = root-> left; delete root; return temp; }// If both children exist else {Node* succParent = root; // Find successor Node *succ = root-> right; while (succ-> left != NULL) { succParent = succ; succ = succ-> left; }// Delete successor.Since successor // is always left child of its parent // we can safely make successor's right // right child as left of its parent. // If there is no succ, then assign // succ-> right to succParent-> right if (succParent != root) succParent-> left = succ-> right; else succParent-> right = succ-> right; // Copy Successor Data to root root-> key = succ-> key; // Delete Successor and return root delete succ; return root; } }// Driver Program to test above functions int main() { /* Let us create following BST 50 /\ 3070 /\/\ 20406080 */ 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); printf ( "\nDelete 20\n" ); root = deleteNode(root, 20); printf ( "Inorder traversal of the modified tree \n" ); inorder(root); printf ( "\nDelete 30\n" ); root = deleteNode(root, 30); printf ( "Inorder traversal of the modified tree \n" ); inorder(root); printf ( "\nDelete 50\n" ); root = deleteNode(root, 50); printf ( "Inorder traversal of the modified tree \n" ); inorder(root); return 0; }

输出如下：

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

谢谢Wolffgang010用于建议上述优化。
相关链接：

• 二叉搜索树介绍, 搜索和插入/ a>
• 二叉搜索树测验
• BST的编码实践
• 关于BST的所有文章

如果发现任何不正确的地方, 或者想分享有关上述主题的更多信息, 请发表评论。

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