Insert Operation

To insert an element:

$1.$ Add it at the end of the array (next position in the complete tree).

$2.$ "Bubble up": while it's smaller than its parent (in a min-heap), swap with parent.

$3.$ Stop when heap property is restored.

function insert(heap, value)
    append value to heap
    i := length of heap - 1
    while i > 0 and heap[i] < heap[parent(i)]
        swap heap[i] and heap[parent(i)]
        i := parent(i)

At most $O(\log n)$ swaps, one per level.