Binary heap — the array index math for parent and children.

A binary heap is a complete tree, so it lives in a flat array and the parent/child links are pure arithmetic — no node objects, no pointers. For a node at index i (zero-based):

int parent(int i) { return (i - 1) / 2; }
int left(int i)   { return 2 * i + 1; }
int right(int i)  { return 2 * i + 2; }

Why these formulas

Level d of a complete tree holds 2^d nodes laid out contiguously. Walking left-to-right, top-to-bottom, the children of node i always land at 2i + 1 and 2i + 2, and the inverse of either maps back through (i - 1) / 2 thanks to integer truncation: (2i+1 - 1)/2 = i and (2i+2 - 1)/2 = i.

         0
      / \
     1   2
    / \  / \
   3  4 5
   
parent(4) = (4-1)/2 = 1
left(1)   = 2*1+1   = 3
right(1)  = 2*1+2   = 4

Practical details

• Bounds: index i is internal (has at least one child) iff 2i + 1 < n. The last internal node is (n / 2) - 1 — useful for build-heap and for the heapify loop.
• One-based variant: some textbooks store from index 1, giving the cleaner parent = i/2, left = 2i, right = 2i + 1. Java's PriorityQueue uses the zero-based form above.
• No gaps: completeness is what guarantees the array is dense. The moment you allow holes, the index math breaks — which is why insert appends at the end and delete fills from the end.