9.3.1. Valid Heap Example 1

So, let us start looking first at what a heap test. So, a binary tree is a tree where we have a root and every node has 0, 1 or 2 children. So, binary trees in general can have arbitrary shapes. So, we could have binary trees which look like this, where the root has 1 child, this has 2 children or it could look even more skewed in one direction. So, binary trees can have very strange shapes, but a heap is a binary tree which has a very specific shape, where we fill up the tree nodes or we add the tree nodes in a specific order.

So, first we start at the root, then we must add the left child of the root, then the right child and this way keep going level by level left to right. So, we add this node, then we add this node, then we add this node. So, once I know how many nodes are there in the tree, I know precisely what the shape is, so the shape is fixed. So, that is the first feature of the heap that if I have a heap with n nodes, then the shape of the tree is deterministically fixed by this rule that the n nodes must be inserted top to bottom, left to right.

A heap is a specialized type of binary tree. The main characteristic of a binary tree is that every node can have either 0, 1, or 2 children. In contrast, heaps must follow a strict structural rule: they are filled level by level, starting from the root, then moving to the left child, followed by the right child, and continuing this pattern until all nodes are added. This ensures that for a heap with 'n' nodes, the overall shape is always the same and is predictable based on the number of nodes.

Think of a heap as a packed box in which you are placing books. You fill the box from the bottom layer, moving left to right, and only filling up one layer completely before going on to the next layer. This creates a stack of books that is easy to manage and clearly visible with a defined order.

So, we have a value property. So, the value property says that... So, what does happening in the tree is that we have nodes and each node is the value, so whenever I see a node with value v1 which has children v2 and v3, then what we want is, this is bigger than or equal to v2 and bigger than or equal to v3. So, among these three nodes the largest one must be v1, so this is what is called the max heap property. So, this is the local property, it just tells us at every node look at that node, look at the 2 children, the node must be bigger than its 2 children.

The heap also has a value property, specifically known as the max heap property. This means that for any given node, the value of that node must be greater than or equal to the values of its child nodes. For instance, if a parent node has a value of 'v1' and two children with values 'v2' and 'v3', then 'v1' must be larger than both 'v2' and 'v3'. This ensures that the largest element is always at the root of the heap, which is essential for efficiently locating the highest priority item in a priority queue.

You can think of this property as being similar to the hierarchy in a company. Imagine the CEO (the root) must have a higher status or authority than the department heads (the child nodes). This structure ensures clarity in organization — just as in a heap where the highest value node (the one with the highest task priority) must always be at the top.

So here is an example of the heap with 4 nodes, so first because it is 4 nodes, every 4 node heap will have the shape. Because, the first node will be the root, the second will be the root's left child, third node will be the right child and the fourth node will start a new line, then moreover we can check the heap property. So, we see the 24 is bigger than 11, 24 is bigger than 7. So, this is a valid node for a heap property, 11 is bigger than 10 there is no right child, so this is a valid heap, there is no child of 7 at all.

In the example given, a heap is shown with four nodes. The structure adheres to the heap's rules, with the nodes strategically placed to maintain the shape. The values of the nodes follow the max heap property: the root node (24) is larger than both of its children (11 and 7). This configuration is valid because all nodes adhere to the required properties for a max heap.

Imagine a pyramid where the largest person stands at the top (the root), and everyone else (the children) must be smaller than the one on top. This ensures that the person at the peak has the highest rank amongst all, just like the largest value in a heap is always positioned at the top.

So, now we have to implement these two operations on heaps, insert and delete max. So, let us see how it works? So, first let us insert 12, so insert 12 means I have to add a value to the heap. So, the first thing when I add a value to the heap is I must expand a tree. So, where do I put this node? So, this now fixed because we know that heaps can only grow and shrink in a particular way, so I must add the new node left to right, top to bottom. So, in this case if I go left to right, top to bottom I come to this point, now I cannot add anything more, so it must come at the next level.

When inserting a new value into a heap, the new node must be added in a way that maintains both the shape and the property of the heap. This means that it must be added in a left-to-right, top-to-bottom manner. If the new value disrupts the max heap property, it may require further adjustments (swaps) to maintain the integrity of the heap after insertion.

Think of adding a new piece to a puzzle. Each piece must fit in its slot (in this case, according to the existing structure), and if it doesn't fit the overall picture (heap property), you need to swap or adjust neighboring pieces to maintain the final image (the heap structure).

--