建议将此职位作为此职位的先决条件。
八卦树|设置1(搜索)
如前一篇文章所述，Splay树是一种自平衡数据结构，其中最后访问的密钥始终位于根目录。插入操作类似于“二进制搜索树”插入，但要执行其他步骤，以确保新插入的键成为新的根。
以下是在展开树中插入密钥k的不同情况。
1)根为NULL：我们只需分配一个新节点并将其作为根返回即可。
2)展开给定的密钥k。如果k已经存在，则它将成为新的根。如果不存在，则最后访问的叶节点将成为新的根。
3)如果新的根密钥与k相同，则不要执行任何操作，因为k已经存在。
4)否则为新节点分配内存，并将根的密钥与k进行比较。
……。 4.a)如果k小于root的键，则将root设为新节点的右子节点，将root的左子节点复制为新节点的左子节点，并将root的左子节点设为NULL。
……。 4.b)如果k大于root的密钥，则将root设为新节点的左子节点，将root的右子节点复制为新节点的右子节点，并将root的右子节点设为NULL。
5)返回新节点作为树的新根。
例子：
100 [20] 25
/ \ \ / \
50 200 50 20 50
/ insert(25) / \ insert(25) / \
40 ======> 30 100 ========> 30 100
/ 1. Splay(25) \ \ 2. insert 25 \ \
30 40 200 40 200
/
[20] C++
#include
using namespace std;
// An AVL tree node
class node
{
public:
int key;
node *left, *right;
};
/* Helper function that allocates
a new node with the given key and
NULL left and right pointers. */
node* newNode(int key)
{
node* Node = new node();
Node->key = key;
Node->left = Node->right = NULL;
return (Node);
}
// A utility function to right
// rotate subtree rooted with y
// See the diagram given above.
node *rightRotate(node *x)
{
node *y = x->left;
x->left = y->right;
y->right = x;
return y;
}
// A utility function to left
// rotate subtree rooted with x
// See the diagram given above.
node *leftRotate(node *x)
{
node *y = x->right;
x->right = y->left;
y->left = x;
return y;
}
// This function brings the key at
// root if key is present in tree.
// If key is not present, then it
// brings the last accessed item at
// root. This function modifies the
// tree and returns the new root
node *splay(node *root, int key)
{
// Base cases: root is NULL or
// key is present at root
if (root == NULL || root->key == key)
return root;
// Key lies in left subtree
if (root->key > key)
{
// Key is not in tree, we are done
if (root->left == NULL) return root;
// Zig-Zig (Left Left)
if (root->left->key > key)
{
// First recursively bring the
// key as root of left-left
root->left->left = splay(root->left->left, key);
// Do first rotation for root,
// second rotation is done after else
root = rightRotate(root);
}
else if (root->left->key < key) // Zig-Zag (Left Right)
{
// First recursively bring
// the key as root of left-right
root->left->right = splay(root->left->right, key);
// Do first rotation for root->left
if (root->left->right != NULL)
root->left = leftRotate(root->left);
}
// Do second rotation for root
return (root->left == NULL)? root: rightRotate(root);
}
else // Key lies in right subtree
{
// Key is not in tree, we are done
if (root->right == NULL) return root;
// Zig-Zag (Right Left)
if (root->right->key > key)
{
// Bring the key as root of right-left
root->right->left = splay(root->right->left, key);
// Do first rotation for root->right
if (root->right->left != NULL)
root->right = rightRotate(root->right);
}
else if (root->right->key < key)// Zag-Zag (Right Right)
{
// Bring the key as root of
// right-right and do first rotation
root->right->right = splay(root->right->right, key);
root = leftRotate(root);
}
// Do second rotation for root
return (root->right == NULL)? root: leftRotate(root);
}
}
// Function to insert a new key k
// in splay tree with given root
node *insert(node *root, int k)
{
// Simple Case: If tree is empty
if (root == NULL) return newNode(k);
// Bring the closest leaf node to root
root = splay(root, k);
// If key is already present, then return
if (root->key == k) return root;
// Otherwise allocate memory for new node
node *newnode = newNode(k);
// If root's key is greater, make
// root as right child of newnode
// and copy the left child of root to newnode
if (root->key > k)
{
newnode->right = root;
newnode->left = root->left;
root->left = NULL;
}
// If root's key is smaller, make
// root as left child of newnode
// and copy the right child of root to newnode
else
{
newnode->left = root;
newnode->right = root->right;
root->right = NULL;
}
return newnode; // newnode becomes new root
}
// A utility function to print
// preorder traversal of the tree.
// The function also prints height of every node
void preOrder(node *root)
{
if (root != NULL)
{
cout<key<<" ";
preOrder(root->left);
preOrder(root->right);
}
}
/* Driver code*/
int main()
{
node *root = newNode(100);
root->left = newNode(50);
root->right = newNode(200);
root->left->left = newNode(40);
root->left->left->left = newNode(30);
root->left->left->left->left = newNode(20);
root = insert(root, 25);
cout<<"Preorder traversal of the modified Splay tree is \n";
preOrder(root);
return 0;
}
// This code is contributed by rathbhupendra C
// This code is adopted from http://algs4.cs.princeton.edu/33balanced/SplayBST.java.html
#include
#include
// An AVL tree node
struct node
{
int key;
struct node *left, *right;
};
/* Helper function that allocates a new node with the given key and
NULL left and right pointers. */
struct node* newNode(int key)
{
struct node* node = (struct node*)malloc(sizeof(struct node));
node->key = key;
node->left = node->right = NULL;
return (node);
}
// A utility function to right rotate subtree rooted with y
// See the diagram given above.
struct node *rightRotate(struct node *x)
{
struct node *y = x->left;
x->left = y->right;
y->right = x;
return y;
}
// A utility function to left rotate subtree rooted with x
// See the diagram given above.
struct node *leftRotate(struct node *x)
{
struct node *y = x->right;
x->right = y->left;
y->left = x;
return y;
}
// This function brings the key at root if key is present in tree.
// If key is not present, then it brings the last accessed item at
// root. This function modifies the tree and returns the new root
struct node *splay(struct node *root, int key)
{
// Base cases: root is NULL or key is present at root
if (root == NULL || root->key == key)
return root;
// Key lies in left subtree
if (root->key > key)
{
// Key is not in tree, we are done
if (root->left == NULL) return root;
// Zig-Zig (Left Left)
if (root->left->key > key)
{
// First recursively bring the key as root of left-left
root->left->left = splay(root->left->left, key);
// Do first rotation for root, second rotation is done after else
root = rightRotate(root);
}
else if (root->left->key < key) // Zig-Zag (Left Right)
{
// First recursively bring the key as root of left-right
root->left->right = splay(root->left->right, key);
// Do first rotation for root->left
if (root->left->right != NULL)
root->left = leftRotate(root->left);
}
// Do second rotation for root
return (root->left == NULL)? root: rightRotate(root);
}
else // Key lies in right subtree
{
// Key is not in tree, we are done
if (root->right == NULL) return root;
// Zig-Zag (Right Left)
if (root->right->key > key)
{
// Bring the key as root of right-left
root->right->left = splay(root->right->left, key);
// Do first rotation for root->right
if (root->right->left != NULL)
root->right = rightRotate(root->right);
}
else if (root->right->key < key)// Zag-Zag (Right Right)
{
// Bring the key as root of right-right and do first rotation
root->right->right = splay(root->right->right, key);
root = leftRotate(root);
}
// Do second rotation for root
return (root->right == NULL)? root: leftRotate(root);
}
}
// Function to insert a new key k in splay tree with given root
struct node *insert(struct node *root, int k)
{
// Simple Case: If tree is empty
if (root == NULL) return newNode(k);
// Bring the closest leaf node to root
root = splay(root, k);
// If key is already present, then return
if (root->key == k) return root;
// Otherwise allocate memory for new node
struct node *newnode = newNode(k);
// If root's key is greater, make root as right child
// of newnode and copy the left child of root to newnode
if (root->key > k)
{
newnode->right = root;
newnode->left = root->left;
root->left = NULL;
}
// If root's key is smaller, make root as left child
// of newnode and copy the right child of root to newnode
else
{
newnode->left = root;
newnode->right = root->right;
root->right = NULL;
}
return newnode; // newnode becomes new root
}
// A utility function to print preorder traversal of the tree.
// The function also prints height of every node
void preOrder(struct node *root)
{
if (root != NULL)
{
printf("%d ", root->key);
preOrder(root->left);
preOrder(root->right);
}
}
/* Driver program to test above function*/
int main()
{
struct node *root = newNode(100);
root->left = newNode(50);
root->right = newNode(200);
root->left->left = newNode(40);
root->left->left->left = newNode(30);
root->left->left->left->left = newNode(20);
root = insert(root, 25);
printf("Preorder traversal of the modified Splay tree is \n");
preOrder(root);
return 0;
} Java
import java.util.*;
class GFG{
// An AVL tree node
static class node
{
int key;
node left, right;
};
/* Helper function that allocates
a new node with the given key and
null left and right pointers. */
static node newNode(int key)
{
node Node = new node();
Node.key = key;
Node.left = Node.right = null;
return (Node);
}
// A utility function to right
// rotate subtree rooted with y
// See the diagram given above.
static node rightRotate(node x)
{
node y = x.left;
x.left = y.right;
y.right = x;
return y;
}
// A utility function to left
// rotate subtree rooted with x
// See the diagram given above.
static node leftRotate(node x)
{
node y = x.right;
x.right = y.left;
y.left = x;
return y;
}
// This function brings the key at
// root if key is present in tree.
// If key is not present, then it
// brings the last accessed item at
// root. This function modifies the
// tree and returns the new root
static node splay(node root, int key)
{
// Base cases: root is null or
// key is present at root
if (root == null || root.key == key)
return root;
// Key lies in left subtree
if (root.key > key)
{
// Key is not in tree, we are done
if (root.left == null) return root;
// Zig-Zig (Left Left)
if (root.left.key > key)
{
// First recursively bring the
// key as root of left-left
root.left.left = splay(root.left.left, key);
// Do first rotation for root,
// second rotation is done after else
root = rightRotate(root);
}
else if (root.left.key < key) // Zig-Zag (Left Right)
{
// First recursively bring
// the key as root of left-right
root.left.right = splay(root.left.right, key);
// Do first rotation for root.left
if (root.left.right != null)
root.left = leftRotate(root.left);
}
// Do second rotation for root
return (root.left == null)? root: rightRotate(root);
}
else // Key lies in right subtree
{
// Key is not in tree, we are done
if (root.right == null) return root;
// Zig-Zag (Right Left)
if (root.right.key > key)
{
// Bring the key as root of right-left
root.right.left = splay(root.right.left, key);
// Do first rotation for root.right
if (root.right.left != null)
root.right = rightRotate(root.right);
}
else if (root.right.key < key)// Zag-Zag (Right Right)
{
// Bring the key as root of
// right-right and do first rotation
root.right.right = splay(root.right.right, key);
root = leftRotate(root);
}
// Do second rotation for root
return (root.right == null)? root: leftRotate(root);
}
}
// Function to insert a new key k
// in splay tree with given root
static node insert(node root, int k)
{
// Simple Case: If tree is empty
if (root == null) return newNode(k);
// Bring the closest leaf node to root
root = splay(root, k);
// If key is already present, then return
if (root.key == k) return root;
// Otherwise allocate memory for new node
node newnode = newNode(k);
// If root's key is greater, make
// root as right child of newnode
// and copy the left child of root to newnode
if (root.key > k)
{
newnode.right = root;
newnode.left = root.left;
root.left = null;
}
// If root's key is smaller, make
// root as left child of newnode
// and copy the right child of root to newnode
else
{
newnode.left = root;
newnode.right = root.right;
root.right = null;
}
return newnode; // newnode becomes new root
}
// A utility function to print
// preorder traversal of the tree.
// The function also prints height of every node
static void preOrder(node root)
{
if (root != null)
{
System.out.print(root.key+" ");
preOrder(root.left);
preOrder(root.right);
}
}
/* Driver code*/
public static void main(String[] args)
{
node root = newNode(100);
root.left = newNode(50);
root.right = newNode(200);
root.left.left = newNode(40);
root.left.left.left = newNode(30);
root.left.left.left.left = newNode(20);
root = insert(root, 25);
System.out.print("Preorder traversal of the modified Splay tree is \n");
preOrder(root);
}
}
// This code is contributed by Rajput-Ji输出：
Preorder traversal of the modified Splay tree is
25 20 50 30 40 100 200