The binary search algorithm and the binary search tree (BST) are two distinct concepts that, while related, serve different purposes in computerÂ science.

The binary search algorithm is a searching technique used on sorted arrays or containers. It operates by repeatedly dividing the search interval inÂ half.

If the value of the search key is less than the item in the middle of the interval, the algorithm narrows the interval to the lowerÂ half.

Otherwise, it narrows it to the upper half. This process continues until the search key is found or the interval isÂ empty.

**Key Characteristics**:

**Requires Sorted Data**: Binary search can only be applied to sorted arrays or containers.**Time Complexity**: $O(log n)$, where n is the number of elements.**Space Complexity**: $O(1)$, as it doesn't require additional space.

The following program uses the standard libraryâ€™s binary searchÂ algorithm:

```
#include <algorithm>
#include <iostream>
#include <vector>
int main() {
std::vector<int> Numbers{1, 2, 3, 4, 5};
int target = 4;
bool found = std::binary_search(
Numbers.begin(), Numbers.end(), target);
std::cout << "The number " << target
<< (found ? " was" : " was not") << " found";
}
```

`The number 4 was found`

A binary search tree is a data structure that facilitates fast lookup, addition, and removal of items. Each node in the BST has up to two children, referred to as the left child and the rightÂ child.

For any node, the left subtree contains values less than the node's value, and the right subtree contains values greater than the node'sÂ value.

**Key Characteristics**:

**Dynamic**: BSTs can grow and shrink dynamically, unlike arrays.**Time Complexity**: $O(log n)$ on average for insertion, deletion, and search operations. However, in the worst case (e.g., a completely unbalanced tree), the time complexity can degrade to $O(n)$.**Space Complexity**: $O(n)$, where n is the number of nodes.

The following program creates a binary search tree fromÂ scratch:

```
#include <iostream>
struct Node {
int data;
Node* left;
Node* right;
};
Node* newNode(int data) {
Node* node = new Node();
node->data = data;
node->left = nullptr;
node->right = nullptr;
return node;
}
Node* insert(Node* root, int data) {
if (root == nullptr) {
return newNode(data);
}
if (data < root->data) {
root->left = insert(root->left, data);
} else if (data > root->data) {
root->right = insert(root->right, data);
}
return root;
}
bool search(Node* root, int data) {
if (root == nullptr) {
return false;
}
if (root->data == data) {
return true;
}
if (data < root->data) {
return search(root->left, data);
} else {
return search(root->right, data);
}
}
int main() {
Node* root = nullptr;
root = insert(root, 4);
insert(root, 2);
insert(root, 5);
insert(root, 1);
insert(root, 3);
int target = 4;
bool found = search(root, target);
std::cout << "The number " << target
<< (found ? " was" : " was not") << " found";
}
```

`The number 4 was found`

**Binary Search Algorithm**: A technique for finding an item in a sorted array with $O(log n)$ time complexity.**Binary Search Tree**: A dynamic data structure that supports fast insertion, deletion, and lookup operations, with $O(log n)$ average time complexity.

Understanding both concepts is crucial for efficient searching and data management in variousÂ applications.

Answers to questions are automatically generated and may not have been reviewed.

This Question is from the Lesson:### Binary Search in C++

An introduction to the advantages of binary search, and how to use it with the C++ standard library algorithms `binary_search()`

, `lower_bound()`

, `upper_bound()`

, and `equal_range()`