Nodes in a heap are identified by their serial numbers. Let us assume that we have a heap in which nodes can have up to k children. The serial number of the root is defined as 1. Let us number children of a node as 0,1,…k − 1. Then child i of a node with a serial number of n is defined as k ⋅ n + i.
Given that a node has a serial number of n, then the serial number of its parent is ⌊n∕k⌋ (the floor of n divided by k).
Combined with the “fill children from left to right” requirement in the previous section, this means node k ⋅n + i must be filled before node k ⋅ n + (i + 1) for some i ∈ [0…n − 2].
Let us assume the leaf with the least serial number is i If there are exactly m leaves in the heap, then one can prove that all the nodes have serial numbers j ∈ [i…i + m − 1].