The heap data structure is a fundamental concept in computer science and is widely used in many applications. In this article, we will explore what the heap data structure is, its properties, and how it can be implemented with an example.

## What Is a Heap?

A heap is a complete binary tree that satisfies the **heap property**. The heap property states that for every node in the tree, the value of that node is greater than or equal to (in a max heap) or less than or equal to (in a min heap) the values of its children.

The heap property can be visualized as a specific ordering of elements in a binary tree. In a max heap, the parent nodes have values greater than or equal to their children. Conversely, in a min heap, parent nodes have values less than or equal to their children.

## Heap Operations

There are two main operations performed on heaps:

**Insertion:**Adding an element to the heap.**Deletion:**Removing an element from the heap.

## Example: Max Heap

To better understand how heaps work, let’s look at an example of a max heap. Consider the following array of numbers: [9, 7, 6, 4, 3, 1]. We can build a max heap from this array by following these steps:

- Create an empty binary tree.
- Insert each element from the array into the binary tree one by one.
- If inserting an element violates the heap property, swap it with its parent until the property is satisfied.

Let’s go through the example step by step:

### Step 1:

Create an empty binary tree.

### Step 2:

Insert the first element, 9, into the binary tree. Since it is the only element, it satisfies the heap property.

### Step 3:

Insert the second element, 7. Since 7 is smaller than its parent (9), we swap them to satisfy the heap property.

### Step 4:

Insert the third element, 6. It is smaller than its parent (9), so we swap them. However, it is now greater than its new parent (7), so another swap is needed.

### Step 5:

Insert the fourth element, 4. We also need to perform additional swaps to satisfy the heap property.

### Step 6:

Insert the fifth element, 3. We continue swapping until the heap property is satisfied.

### Step 7:

Insert the last element, 1. Similarly, we swap it with its parent until the heap property holds for all nodes.

The resulting max heap from our example array [9, 7, 6, 4, 3, 1] will look like this:

9/ \76/ \ /431

As we can see, the heap property is satisfied for every node in the tree.

## Conclusion

The heap data structure is a powerful tool that allows efficient insertion and deletion of elements. It is commonly used in priority queues, sorting algorithms like heapsort, and graph algorithms such as Dijkstra’s algorithm. Understanding heaps and their properties is crucial for any programmer or computer science enthusiast.

By now, you should have a good understanding of what a heap data structure is and how it can be implemented with an example. Remember to always consider the heap property when working with heaps to ensure their correctness.