二进制搜索树|套装2(删除)

我们已经讨论了BST搜索和插入操作。在这篇文章中，讨论了删除操作。当我们删除节点时，会出现三种可能性。
1)要删除的节点是叶子：只需从树中删除即可。

50                            50
           /     \         delete(20)      /   \
          30      70       --------->    30     70 
         /  \    /  \                     \    /  \ 
       20   40  60   80                   40  60   80

2)要删除的节点只有一个子节点：将子节点复制到该节点并删除该子节点

50                            50
           /     \         delete(30)      /   \
          30      70       --------->    40     70 
            \    /  \                          /  \ 
            40  60   80                       60   80

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。

50                            60
           /     \         delete(50)      /   \
          40      70       --------->    40    70 
                 /  \                            \ 
                60   80                           80

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。

Python3

// C++ program to demonstrate
// delete operation in binary
// search tree
#include 
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 has no child
        if (root.left==NULL and root.right==NULL):
            return NULL
       
        // node with only one child or no child
        elif (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 Code
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);
 
    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 
#include 
 
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 Code
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);
 
    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
      delete an existing 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 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);
 
        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 in  left 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 code
""" 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)
 
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
      delete an existing 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：

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：

时间复杂度：删除操作的最坏情况下的时间复杂度为O(h)，其中h是二进制搜索树的高度。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)

对上述两个子案例的代码进行了优化：
在上面的递归代码中，我们递归地调用后继者的delete()。我们可以通过跟踪后继者的父节点来避免递归调用，这样我们就可以通过使父代的子代为NULL来简单地删除后继者。我们知道后继者将始终是叶节点。

C++

// C++ program to implement optimized delete in BST.
#include 
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 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);
 
    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;
}

Python3

# Python3 program to implement
# optimized delete in BST.
 
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, 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 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 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
 
    # Recursive calls for ancestors of
    # node to be deleted
    if key < root.key:
        root.left = deleteNode(root.left, key)
        return root
 
    elif(key > root.key):
        root.right = deleteNode(root.right, key)
        return root
 
    # We reach here when root is the node
    # to be deleted.
     
    # If root node is a leaf node
     
    if root.left is None and root.right is None:
          return None
 
    # If one of the children is empty
 
    if root.left is None:
        temp = root.right
        root = None
        return temp
 
    elif root.right is None:
        temp = root.left
        root = None
        return temp
 
    # If both children exist
 
    succParent = root
 
    # Find Successor
 
    succ = root.right
 
    while succ.left != None:
        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
 
    return root
 
 
# Driver code
""" 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)
 
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 Shivam Bhat (shivambhat45)

输出

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 

感谢wolffgang010建议上述优化。
相关链接：

• 二进制搜索树介绍，搜索和插入/ a>
• 二进制搜索树上的测验
• BST的编码实践
• 关于BST的所有文章