Final answer:
The third smallest element in a binary heap could be among the children or grandchildren of the root node. It is not bound to be the root node itself, the last node, or the third node before the last.
Step-by-step explanation:
In a binary heap, which is a type of complete binary tree, the properties dictate that each parent node is less than or equal to (in a min-heap) or greater than or equal to (in a max-heap) its child nodes. Given that a heap must be a complete binary tree, the nodes are added level by level, left to right. With this in mind, the third smallest element in a min-heap of more than three nodes would not necessarily be the root node, which is the smallest. Similarly, it is not guaranteed to always be the last node or the third node before the last, as the heap property does not require siblings to be in sorted order.
However, the third smallest element could be either one of the children of the root node (if both children are larger than the root but still among the smallest in the heap) or one of the grandchildren of the root node. The answer to where the third smallest key might be in a binary heap tree is thus:
- a. One of the grandchildren of the root node
- e. One of the children of the root node