How quickly (in terms of comparisons) can you find the path to a leaf down which the element E placed in the root will travel? Note that for now you do not have to do any swapping; you only have to identify the path.
Assume that we store the heap in an array and that the leaf in question is stored in entry x. Which entries in the list does the path mentioned above correspond to, expressed in terms of x?
Can you now find the correct location for E faster?
How quickly (in terms of comparisons between elements) can you insert the element into the correct location? What is the total number of comparison for the entire operation?