Dado un árbol binario, una ruta completa se define como una ruta desde la raíz hasta una hoja. La suma de todos los Nodes en ese camino se define como la suma de ese camino. Dado un número K, debe eliminar (podar el árbol) todos los Nodes que no se encuentran en ningún camino con sum>=k.
Nota: un Node puede ser parte de varias rutas. Por lo tanto, debemos eliminarlo solo en caso de que todas las rutas tengan una suma menor que K.
C++
#include <bits/stdc++.h>
using namespace std;
// A utility function to get maximum of two integers
int max(int l, int r) { return (l > r ? l : r); }
// A Binary Tree Node
struct Node
{
int data;
struct Node *left, *right;
};
// A utility function to create a new Binary Tree node with given data
struct Node* newNode(int data)
{
struct Node* node = (struct Node*) malloc(sizeof(struct Node));
node->data = data;
node->left = node->right = NULL;
return node;
}
// print the tree in LVR (Inorder traversal) way.
void print(struct Node *root)
{
if (root != NULL)
{
print(root->left);
cout <<" "<< root->data;
print(root->right);
}
}
/* Main function which truncates the binary tree. */
struct Node *pruneUtil(struct Node *root, int k, int *sum)
{
// Base Case
if (root == NULL) return NULL;
// Initialize left and right sums as sum from root to
// this node (including this node)
int lsum = *sum + (root->data);
int rsum = lsum;
// Recursively prune left and right subtrees
root->left = pruneUtil(root->left, k, &lsum);
root->right = pruneUtil(root->right, k, &rsum);
// Get the maximum of left and right sums
*sum = max(lsum, rsum);
// If maximum is smaller than k, then this node
// must be deleted
if (*sum < k)
{
free(root);
root = NULL;
}
return root;
}
// A wrapper over pruneUtil()
struct Node *prune(struct Node *root, int k)
{
int sum = 0;
return pruneUtil(root, k, &sum);
}
// Driver program to test above function
int main()
{
int k = 45;
struct Node *root = newNode(1);
root->left = newNode(2);
root->right = newNode(3);
root->left->left = newNode(4);
root->left->right = newNode(5);
root->right->left = newNode(6);
root->right->right = newNode(7);
root->left->left->left = newNode(8);
root->left->left->right = newNode(9);
root->left->right->left = newNode(12);
root->right->right->left = newNode(10);
root->right->right->left->right = newNode(11);
root->left->left->right->left = newNode(13);
root->left->left->right->right = newNode(14);
root->left->left->right->right->left = newNode(15);
cout <<"Tree before truncation\n";
print(root);
root = prune(root, k); // k is 45
cout <<"\n\nTree after truncation\n";
print(root);
return 0;
}
// This code is contributed by shivanisinghss2110
C
#include <stdio.h>
#include <stdlib.h>
// A utility function to get maximum of two integers
int max(int l, int r) { return (l > r ? l : r); }
// A Binary Tree Node
struct Node
{
int data;
struct Node *left, *right;
};
// A utility function to create a new Binary Tree node with given data
struct Node* newNode(int data)
{
struct Node* node = (struct Node*) malloc(sizeof(struct Node));
node->data = data;
node->left = node->right = NULL;
return node;
}
// print the tree in LVR (Inorder traversal) way.
void print(struct Node *root)
{
if (root != NULL)
{
print(root->left);
printf("%d ",root->data);
print(root->right);
}
}
/* Main function which truncates the binary tree. */
struct Node *pruneUtil(struct Node *root, int k, int *sum)
{
// Base Case
if (root == NULL) return NULL;
// Initialize left and right sums as sum from root to
// this node (including this node)
int lsum = *sum + (root->data);
int rsum = lsum;
// Recursively prune left and right subtrees
root->left = pruneUtil(root->left, k, &lsum);
root->right = pruneUtil(root->right, k, &rsum);
// Get the maximum of left and right sums
*sum = max(lsum, rsum);
// If maximum is smaller than k, then this node
// must be deleted
if (*sum < k)
{
free(root);
root = NULL;
}
return root;
}
// A wrapper over pruneUtil()
struct Node *prune(struct Node *root, int k)
{
int sum = 0;
return pruneUtil(root, k, &sum);
}
// Driver program to test above function
int main()
{
int k = 45;
struct Node *root = newNode(1);
root->left = newNode(2);
root->right = newNode(3);
root->left->left = newNode(4);
root->left->right = newNode(5);
root->right->left = newNode(6);
root->right->right = newNode(7);
root->left->left->left = newNode(8);
root->left->left->right = newNode(9);
root->left->right->left = newNode(12);
root->right->right->left = newNode(10);
root->right->right->left->right = newNode(11);
root->left->left->right->left = newNode(13);
root->left->left->right->right = newNode(14);
root->left->left->right->right->left = newNode(15);
printf("Tree before truncation\n");
print(root);
root = prune(root, k); // k is 45
printf("\n\nTree after truncation\n");
print(root);
return 0;
}
Java
// Java program to implement
// the above approach
import java.util.*;
class GFG
{
// A utility function to get
// maximum of two integers
static int max(int l, int r)
{
return (l > r ? l : r);
}
// A Binary Tree Node
static class Node
{
int data;
Node left, right;
};
static class INT
{
int v;
INT(int a)
{
v = a;
}
}
// A utility function to create
// a new Binary Tree node with
// given data
static Node newNode(int data)
{
Node node = new Node();
node.data = data;
node.left = node.right = null;
return node;
}
// print the tree in LVR
// (Inorder traversal) way.
static void print(Node root)
{
if (root != null)
{
print(root.left);
System.out.print(root.data + " ");
print(root.right);
}
}
// Main function which
// truncates the binary tree.
static Node pruneUtil(Node root, int k,
INT sum)
{
// Base Case
if (root == null) return null;
// Initialize left and right
// sums as sum from root to
// this node (including this node)
INT lsum = new INT(sum.v + (root.data));
INT rsum = new INT(lsum.v);
// Recursively prune left
// and right subtrees
root.left = pruneUtil(root.left, k, lsum);
root.right = pruneUtil(root.right, k, rsum);
// Get the maximum of
// left and right sums
sum.v = max(lsum.v, rsum.v);
// If maximum is smaller
// than k, then this node
// must be deleted
if (sum.v < k)
{
root = null;
}
return root;
}
// A wrapper over pruneUtil()
static Node prune(Node root, int k)
{
INT sum = new INT(0);
return pruneUtil(root, k, sum);
}
// Driver Code
public static void main(String args[])
{
int k = 45;
Node root = newNode(1);
root.left = newNode(2);
root.right = newNode(3);
root.left.left = newNode(4);
root.left.right = newNode(5);
root.right.left = newNode(6);
root.right.right = newNode(7);
root.left.left.left = newNode(8);
root.left.left.right = newNode(9);
root.left.right.left = newNode(12);
root.right.right.left = newNode(10);
root.right.right.left.right = newNode(11);
root.left.left.right.left = newNode(13);
root.left.left.right.right = newNode(14);
root.left.left.right.right.left = newNode(15);
System.out.println("Tree before truncation\n");
print(root);
root = prune(root, k); // k is 45
System.out.println("\n\nTree after truncation\n");
print(root);
}
}
// This code is contributed by Arnab Kundu
Python3
# A class to create a new Binary Tree
# node with given data
class newNode:
def __init__(self, data):
self.data = data
self.left = self.right = None
# print the tree in LVR (Inorder traversal) way.
def Print(root):
if (root != None):
Print(root.left)
print(root.data, end = " ")
Print(root.right)
# Main function which truncates
# the binary tree.
def pruneUtil(root, k, Sum):
# Base Case
if (root == None):
return None
# Initialize left and right Sums as
# Sum from root to this node
# (including this node)
lSum = [Sum[0] + (root.data)]
rSum = [lSum[0]]
# Recursively prune left and right
# subtrees
root.left = pruneUtil(root.left, k, lSum)
root.right = pruneUtil(root.right, k, rSum)
# Get the maximum of left and right Sums
Sum[0] = max(lSum[0], rSum[0])
# If maximum is smaller than k,
# then this node must be deleted
if (Sum[0] < k[0]):
root = None
return root
# A wrapper over pruneUtil()
def prune(root, k):
Sum = [0]
return pruneUtil(root, k, Sum)
# Driver Code
if __name__ == '__main__':
k = [45]
root = newNode(1)
root.left = newNode(2)
root.right = newNode(3)
root.left.left = newNode(4)
root.left.right = newNode(5)
root.right.left = newNode(6)
root.right.right = newNode(7)
root.left.left.left = newNode(8)
root.left.left.right = newNode(9)
root.left.right.left = newNode(12)
root.right.right.left = newNode(10)
root.right.right.left.right = newNode(11)
root.left.left.right.left = newNode(13)
root.left.left.right.right = newNode(14)
root.left.left.right.right.left = newNode(15)
print("Tree before truncation")
Print(root)
print()
root = prune(root, k) # k is 45
print("Tree after truncation")
Print(root)
# This code is contributed by PranchalK
C#
using System;
// C# program to implement
// the above approach
public class GFG
{
// A utility function to get
// maximum of two integers
public static int max(int l, int r)
{
return (l > r ? l : r);
}
// A Binary Tree Node
public class Node
{
public int data;
public Node left, right;
}
public class INT
{
public int v;
public INT(int a)
{
v = a;
}
}
// A utility function to create
// a new Binary Tree node with
// given data
public static Node newNode(int data)
{
Node node = new Node();
node.data = data;
node.left = node.right = null;
return node;
}
// print the tree in LVR
// (Inorder traversal) way.
public static void print(Node root)
{
if (root != null)
{
print(root.left);
Console.Write(root.data + " ");
print(root.right);
}
}
// Main function which
// truncates the binary tree.
public static Node pruneUtil(Node root, int k, INT sum)
{
// Base Case
if (root == null)
{
return null;
}
// Initialize left and right
// sums as sum from root to
// this node (including this node)
INT lsum = new INT(sum.v + (root.data));
INT rsum = new INT(lsum.v);
// Recursively prune left
// and right subtrees
root.left = pruneUtil(root.left, k, lsum);
root.right = pruneUtil(root.right, k, rsum);
// Get the maximum of
// left and right sums
sum.v = max(lsum.v, rsum.v);
// If maximum is smaller
// than k, then this node
// must be deleted
if (sum.v < k)
{
root = null;
}
return root;
}
// A wrapper over pruneUtil()
public static Node prune(Node root, int k)
{
INT sum = new INT(0);
return pruneUtil(root, k, sum);
}
// Driver Code
public static void Main(string[] args)
{
int k = 45;
Node root = newNode(1);
root.left = newNode(2);
root.right = newNode(3);
root.left.left = newNode(4);
root.left.right = newNode(5);
root.right.left = newNode(6);
root.right.right = newNode(7);
root.left.left.left = newNode(8);
root.left.left.right = newNode(9);
root.left.right.left = newNode(12);
root.right.right.left = newNode(10);
root.right.right.left.right = newNode(11);
root.left.left.right.left = newNode(13);
root.left.left.right.right = newNode(14);
root.left.left.right.right.left = newNode(15);
Console.WriteLine("Tree before truncation\n");
print(root);
root = prune(root, k); // k is 45
Console.WriteLine("\n\nTree after truncation\n");
print(root);
}
}
// This code is contributed by Shrikant13
Javascript
<script>
// Javascript program to implement
// the above approach
// A utility function to get
// maximum of two integers
function max(l, r)
{
return (l > r ? l : r);
}
class Node
{
constructor(data)
{
this.left = null;
this.right = null;
this.data = data;
}
}
class INT
{
constructor(a)
{
this.v = a;
}
}
// A utility function to create
// a new Binary Tree node with
// given data
function newNode(data)
{
let node = new Node(data);
return node;
}
// print the tree in LVR
// (Inorder traversal) way.
function print(root)
{
if (root != null)
{
print(root.left);
document.write(root.data + " ");
print(root.right);
}
}
// Main function which
// truncates the binary tree.
function pruneUtil(root, k, sum)
{
// Base Case
if (root == null)
return null;
// Initialize left and right
// sums as sum from root to
// this node (including this node)
let lsum = new INT(sum.v + (root.data));
let rsum = new INT(lsum.v);
// Recursively prune left
// and right subtrees
root.left = pruneUtil(root.left, k, lsum);
root.right = pruneUtil(root.right, k, rsum);
// Get the maximum of
// left and right sums
sum.v = max(lsum.v, rsum.v);
// If maximum is smaller
// than k, then this node
// must be deleted
if (sum.v < k)
{
root = null;
}
return root;
}
// A wrapper over pruneUtil()
function prune(root, k)
{
let sum = new INT(0);
return pruneUtil(root, k, sum);
}
// Driver code
let k = 45;
let root = newNode(1);
root.left = newNode(2);
root.right = newNode(3);
root.left.left = newNode(4);
root.left.right = newNode(5);
root.right.left = newNode(6);
root.right.right = newNode(7);
root.left.left.left = newNode(8);
root.left.left.right = newNode(9);
root.left.right.left = newNode(12);
root.right.right.left = newNode(10);
root.right.right.left.right = newNode(11);
root.left.left.right.left = newNode(13);
root.left.left.right.right = newNode(14);
root.left.left.right.right.left = newNode(15);
document.write("Tree before truncation" + "</br>");
print(root);
root = prune(root, k); // k is 45
document.write("</br></br>" +
"Tree after truncation" + "</br>");
print(root);
// This code is contributed by decode2207
</script>
C++
#include <bits/stdc++.h>
using namespace std;
// A Binary Tree Node
struct Node
{
int data;
struct Node *left, *right;
};
// A utility function to create a new Binary
// Tree node with given data
struct Node* newNode(int data)
{
struct Node* node =
(struct Node*) malloc(sizeof(struct Node));
node->data = data;
node->left = node->right = NULL;
return node;
}
// print the tree in LVR (Inorder traversal) way.
void print(struct Node *root)
{
if (root != NULL)
{
print(root->left);
cout << root->data << " ";
print(root->right);
}
}
/* Main function which truncates the binary tree. */
struct Node *prune(struct Node *root, int sum)
{
// Base Case
if (root == NULL) return NULL;
// Recur for left and right subtrees
root->left = prune(root->left, sum - root->data);
root->right = prune(root->right, sum - root->data);
// If we reach leaf whose data is smaller than sum,
// we delete the leaf. An important thing to note
// is a non-leaf node can become leaf when its
// children are deleted.
if (root->left==NULL && root->right==NULL)
{
if (root->data < sum)
{
free(root);
return NULL;
}
}
return root;
}
// Driver program to test above function
int main()
{
int k = 45;
struct Node *root = newNode(1);
root->left = newNode(2);
root->right = newNode(3);
root->left->left = newNode(4);
root->left->right = newNode(5);
root->right->left = newNode(6);
root->right->right = newNode(7);
root->left->left->left = newNode(8);
root->left->left->right = newNode(9);
root->left->right->left = newNode(12);
root->right->right->left = newNode(10);
root->right->right->left->right = newNode(11);
root->left->left->right->left = newNode(13);
root->left->left->right->right = newNode(14);
root->left->left->right->right->left = newNode(15);
cout << "Tree before truncation\n";
print(root);
root = prune(root, k); // k is 45
cout << "\n\nTree after truncation\n";
print(root);
return 0;
}
// This code is contributed
// by Akanksha Rai
C
#include <stdio.h>
#include <stdlib.h>
// A Binary Tree Node
struct Node
{
int data;
struct Node *left, *right;
};
// A utility function to create a new Binary
// Tree node with given data
struct Node* newNode(int data)
{
struct Node* node =
(struct Node*) malloc(sizeof(struct Node));
node->data = data;
node->left = node->right = NULL;
return node;
}
// print the tree in LVR (Inorder traversal) way.
void print(struct Node *root)
{
if (root != NULL)
{
print(root->left);
printf("%d ",root->data);
print(root->right);
}
}
/* Main function which truncates the binary tree. */
struct Node *prune(struct Node *root, int sum)
{
// Base Case
if (root == NULL) return NULL;
// Recur for left and right subtrees
root->left = prune(root->left, sum - root->data);
root->right = prune(root->right, sum - root->data);
// If we reach leaf whose data is smaller than sum,
// we delete the leaf. An important thing to note
// is a non-leaf node can become leaf when its
// children are deleted.
if (root->left==NULL && root->right==NULL)
{
if (root->data < sum)
{
free(root);
return NULL;
}
}
return root;
}
// Driver program to test above function
int main()
{
int k = 45;
struct Node *root = newNode(1);
root->left = newNode(2);
root->right = newNode(3);
root->left->left = newNode(4);
root->left->right = newNode(5);
root->right->left = newNode(6);
root->right->right = newNode(7);
root->left->left->left = newNode(8);
root->left->left->right = newNode(9);
root->left->right->left = newNode(12);
root->right->right->left = newNode(10);
root->right->right->left->right = newNode(11);
root->left->left->right->left = newNode(13);
root->left->left->right->right = newNode(14);
root->left->left->right->right->left = newNode(15);
printf("Tree before truncation\n");
print(root);
root = prune(root, k); // k is 45
printf("\n\nTree after truncation\n");
print(root);
return 0;
}
Java
// Java program to remove all nodes which donot
// lie on path having sum>= k
// Class representing binary tree node
class Node {
int data;
Node left;
Node right;
// Constructor to create a new node
public Node(int data) {
this.data = data;
left = null;
right = null;
}
}
// class to truncate binary tree
class BinaryTree {
Node root;
// recursive method to truncate binary tree
public Node prune(Node root, int sum) {
// base case
if (root == null)
return null;
// recur for left and right subtree
root.left = prune(root.left, sum - root.data);
root.right = prune(root.right, sum - root.data);
// if node is a leaf node whose data is smaller
// than the sum we delete the leaf.An important
// thing to note is a non-leaf node can become
// leaf when its children are deleted.
if (isLeaf(root)) {
if (sum > root.data)
root = null;
}
return root;
}
// utility method to check if node is leaf
public boolean isLeaf(Node root) {
if (root == null)
return false;
if (root.left == null && root.right == null)
return true;
return false;
}
// for print traversal
public void print(Node root) {
// base case
if (root == null)
return;
print(root.left);
System.out.print(root.data + " ");
print(root.right);
}
}
// Driver class to test above function
public class GFG {
public static void main(String args[]) {
BinaryTree tree = new BinaryTree();
tree.root = new Node(1);
tree.root.left = new Node(2);
tree.root.right = new Node(3);
tree.root.left.left = new Node(4);
tree.root.left.right = new Node(5);
tree.root.right.left = new Node(6);
tree.root.right.right = new Node(7);
tree.root.left.left.left = new Node(8);
tree.root.left.left.right = new Node(9);
tree.root.left.right.left = new Node(12);
tree.root.right.right.left = new Node(10);
tree.root.right.right.left.right = new Node(11);
tree.root.left.left.right.left = new Node(13);
tree.root.left.left.right.right = new Node(14);
tree.root.left.left.right.right.left = new Node(15);
System.out.println("Tree before truncation");
tree.print(tree.root);
tree.prune(tree.root, 45);
System.out.println("\nTree after truncation");
tree.print(tree.root);
}
}
// This code is contributed by Shweta Singh
Python3
"""
Python program to remove all nodes which don’t
lie in any path with sum>= k
"""
# binary tree node contains data field , left
# and right pointer
class Node:
# constructor to create tree node
def __init__(self, data):
self.data = data
self.left = None
self.right = None
# Function to remove all nodes which do not
# lie in th sum path
def prune(root, sum):
# Base case
if root is None:
return None
# Recur for left and right subtree
root.left = prune(root.left, sum - root.data)
root.right = prune(root.right, sum - root.data)
# if node is leaf and sum is found greater
# than data than remove node An important
# thing to remember is that a non-leaf node
# can become a leaf when its children are
# removed
if root.left is None and root.right is None:
if sum > root.data:
return None
return root
# inorder traversal
def inorder(root):
if root is None:
return
inorder(root.left)
print(root.data, "", end="")
inorder(root.right)
# Driver program to test above function
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
root.right.left = Node(6)
root.right.right = Node(7)
root.left.left.left = Node(8)
root.left.left.right = Node(9)
root.left.right.left = Node(12)
root.right.right.left = Node(10)
root.right.right.left.right = Node(11)
root.left.left.right.left = Node(13)
root.left.left.right.right = Node(14)
root.left.left.right.right.left = Node(15)
print("Tree before truncation")
inorder(root)
prune(root, 45)
print("\nTree after truncation")
inorder(root)
# This code is contributed by Shweta Singh
C#
using System;
// C# program to remove all nodes which donot
// lie on path having sum>= k
// Class representing binary tree node
public class Node
{
public int data;
public Node left;
public Node right;
// Constructor to create a new node
public Node(int data)
{
this.data = data;
left = null;
right = null;
}
}
// class to truncate binary tree
public class BinaryTree
{
public Node root;
// recursive method to truncate binary tree
public virtual Node prune(Node root, int sum)
{
// base case
if (root == null)
{
return null;
}
// recur for left and right subtree
root.left = prune(root.left, sum - root.data);
root.right = prune(root.right, sum - root.data);
// if node is a leaf node whose data is smaller
// than the sum we delete the leaf.An important
// thing to note is a non-leaf node can become
// leaf when its children are deleted.
if (isLeaf(root))
{
if (sum > root.data)
{
root = null;
}
}
return root;
}
// utility method to check if node is leaf
public virtual bool isLeaf(Node root)
{
if (root == null)
{
return false;
}
if (root.left == null && root.right == null)
{
return true;
}
return false;
}
// for print traversal
public virtual void print(Node root)
{
// base case
if (root == null)
{
return;
}
print(root.left);
Console.Write(root.data + " ");
print(root.right);
}
}
// Driver class to test above function
public class GFG
{
public static void Main(string[] args)
{
BinaryTree tree = new BinaryTree();
tree.root = new Node(1);
tree.root.left = new Node(2);
tree.root.right = new Node(3);
tree.root.left.left = new Node(4);
tree.root.left.right = new Node(5);
tree.root.right.left = new Node(6);
tree.root.right.right = new Node(7);
tree.root.left.left.left = new Node(8);
tree.root.left.left.right = new Node(9);
tree.root.left.right.left = new Node(12);
tree.root.right.right.left = new Node(10);
tree.root.right.right.left.right = new Node(11);
tree.root.left.left.right.left = new Node(13);
tree.root.left.left.right.right = new Node(14);
tree.root.left.left.right.right.left = new Node(15);
Console.WriteLine("Tree before truncation");
tree.print(tree.root);
tree.prune(tree.root, 45);
Console.WriteLine("\nTree after truncation");
tree.print(tree.root);
}
}
// This code is contributed by Shrikant13
Javascript
<script>
// JavaScript program to remove all nodes which donot
// lie on path having sum>= k
// Class representing binary tree node
class Node {
// Constructor to create a new node
constructor(data) {
this.data = data;
this.left = null;
this.right = null;
}
}
// class to truncate binary tree
class BinaryTree {
constructor() {
this.root = null;
}
// recursive method to truncate binary tree
prune(root, sum) {
// base case
if (root == null) {
return null;
}
// recur for left and right subtree
root.left = this.prune(root.left, sum - root.data);
root.right = this.prune(root.right, sum - root.data);
// if node is a leaf node whose data is smaller
// than the sum we delete the leaf.An important
// thing to note is a non-leaf node can become
// leaf when its children are deleted.
if (this.isLeaf(root)) {
if (sum > root.data) {
root = null;
}
}
return root;
}
// utility method to check if node is leaf
isLeaf(root) {
if (root == null) {
return false;
}
if (root.left == null && root.right == null) {
return true;
}
return false;
}
// for print traversal
print(root) {
// base case
if (root == null) {
return;
}
this.print(root.left);
document.write(root.data + " ");
this.print(root.right);
}
}
// Driver class to test above function
var tree = new BinaryTree();
tree.root = new Node(1);
tree.root.left = new Node(2);
tree.root.right = new Node(3);
tree.root.left.left = new Node(4);
tree.root.left.right = new Node(5);
tree.root.right.left = new Node(6);
tree.root.right.right = new Node(7);
tree.root.left.left.left = new Node(8);
tree.root.left.left.right = new Node(9);
tree.root.left.right.left = new Node(12);
tree.root.right.right.left = new Node(10);
tree.root.right.right.left.right = new Node(11);
tree.root.left.left.right.left = new Node(13);
tree.root.left.left.right.right = new Node(14);
tree.root.left.left.right.right.left = new Node(15);
document.write("Tree before truncation <br>");
tree.print(tree.root);
tree.prune(tree.root, 45);
document.write("<br><br>Tree after truncation<br>");
tree.print(tree.root);
</script>
Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA