How does the equation to index non-leaf nodes of a binary tree 'array' in max-heap work for unbalanced binary trees?

Question

According to Wikipedia's entry on binary heaps:

The Build-Max-Heap function that follows, converts an array A which stores a complete binary tree with n nodes to a max-heap by repeatedly using Max-Heapify (down-heapify for a max-heap) in a bottom-up manner. The array elements indexed by floor(n/2) + 1, floor(n/2) + 2, ..., n are all leaves for the tree (assuming that indices start at 1)—thus each is a one-element heap, and does not need to be down-heapified. Build-Max-Heap runs Max-Heapify on each of the remaining tree nodes.

Build-Max-Heap (A): >> for each index i from floor(length(A)/2) down to 1 do: Max-Heapify(A, i)

I also know that to represent a binary tree as an array, you place the root at index 0. Then, left child at indexOfParent*2 + 1 and right child at indexOfParent*2 + 2 is performed for all subsequent nodes

Now, supposing I have a very unbalanced binary tree represented as follows:

 3
 1 8
 45 7
 6
 52
 94

Representing this binary tree as an array with elements (x,n), x is value of a node, n being index of that value within the array, I get arr = [(3,0),(1,1), (8,2), (45,3) (7,6), (6,14),(52,30),(94,62)]

Because of how unbalanced this tree, there are so many chunks in the array with no values. Regardless, we know this array has a length of 62. Back to the formula , n//2 implies 62//2 = 31. Therefore, according to the equation, all values in the array with index < 31 are non-leaf

Yet value 45 at index 3 is very clearly a leaf node. Back to the Wikipedia definition, The array elements indexed by floor(n/2) + 1, floor(n/2) + 2, ..., n are all leaves for the tree

Does this not contradict the value 45 being at index 3 then? I might then try to remove all the 'gaps' in the array but that would represent a completely different binary tree (a balanced one)

So how does the equation for calculating non-leaf nodes work on such unbalanced binary trees?

• Siddhanth Ramakrishnan: Your tree is not a binary heap, so why would you expect that it has that property? Also, the length of the array is 63.