data structures – Why does the formula floor((i-1)/2) find the parent node in a binary heap?

Let’s assume that each tier of the heap is an array.

T1 (                n0                 )

T2 (      n1,                n2        )

T3 (  n3,     n4,       n5,       n6   )

T4 (n7, n8, n9, n10, n11, n12, n13, n14)

It may not be clear, but I want you to imagine that $n0$ is the parent of $n1$ and $n2$.

Likewise, $n1$ is the parent of $n3$ and $n4$.

For example, the second tier’s length is $2$.

i.e. $(n1, n2)$

Let’s say $i$ is the global index of the node in question (i.e. the node’s index if the entire tree were to be collapsed within a single array), and
$j$ is the local index of the node, i.e. the index of the node within its tier.

e.g. In the diagram above, if the node in question is $n3$, $i$ is $3$ and $j$ is $0$, because $n3$ is the first element in the 3rd tier.

i = global index of a node n
j = local index of a node n within the tier where the node exists
T = tier

The maximum number of nodes in a certain tier can be expressed by:

$2^T – 1$ // e.g. when you have 3 tiers, you can have at most 7 nodes, as $2 cdot 2 cdot 2 – 1 = 7$

This is because there are $2^{T-1}$ nodes in each tier (note: tier numbering in our example arbitrarily starts from $1$, not $0$) and the sum of powers of $2$ up to $n$ is equal to $2^{n+1} – 1$, as you can see here https://math.stackexchange.com/questions/1990137/the-idea-behind-the-sum-of-powers-of-2

This means that the global index of the last node in the tier is:

$i_{last} = 2^T-1-1$

While the local index of the last node in the tier is:

$j_{last} = 2^{T-1} – 1$ // since there are $2^{T-1}$ elements in tier $T$

We can now compute the global index of the first node in the tier by subtracting the local index of the last node in the tier from its global index:

$i_{first} = i_{last} – j_{last}$

$i_{first} = 2^T – 1 – 1 – j_{last}$

$i_{first} = 2^T – 1 – 1 – (2^{T-1} – 1)$

$i_{first} = 2^T – 2^{T-1} – 1$

$i_{first} = 2*2^{T-1} -2^{T-1} – 1$

$i_{first} = (2 – 1)*2^{T-1} – 1$

$i_{first} = 2^{T-1} – 1$

We can now compute the global index of a node $n$ by adding its local index and the global index of the first node in its tier:

$i = i_{first} + j$

$i = 2^{T-1} – 1 + j$

Let’s now think about the parent of the node in question.

The parent, n', will be in the previous tier, T-1. 
The indices in T-1 will be referred to as i', j', i'_first ...

Based on what we’ve shown so far, we can say that the global index of the parent node in the previous tier is:

$i_{first}’ = 2^{T-1-1} – 1$

Now, for every 2 predecessor(left sibling) of the node $n$ in tier $T$ there will be 1 predecessor(left sibling) of parent node $n’$ in $T-1$. Also, given index $j$ in $T$, we know there are $j$ predecessors(left sibling) of $n$ in $T$ – i.e. the index of a node is equal to the number of its predecessors(left sibling). So we can conclude that:

$j’ = floor(j/2)$

Putting it all together, we can conclude that

$i’ = i_f’ + j’$

$i’ = 2^{T-1-1} – 1 + floor(j/2)$

Now let’s rework the previous equation for the global index of parent node $n$, $i = 2^{T-1}-1+j$ to the following:

$i + 1 = 2^{T-1} + j$

Finally, let’s compare that to the equation of the global index of the left-child node:

$begin{align}
i’ &= 2^{T-1-1} – 1 + floor(j/2)\
&= 2^{T-1}/2 + floor(j/2) – 1\
&= floor((2^{T-1} + j)/2) – 1\
&= floor((i + 1)/2) – 1 text{//NOTE: Here we use $i + 1 = 2^{T-1} + j$ mentioned above}\
&= floor((i – 1)/2)\
end{align}$