Please prove ths by standard methods.
Let B be the binary representation of the number of elements in a binomial heap. Consider inserting a node in the heap. Prove that, the number of times two trees in the binomial heap are merged into a bigger tree is equal to the number of bit- ips when B is incremented by 1.
and
Consider two binomial heaps H1 and H2, and let the binary representation of the number of elements in the two heaps, be B1 and B2 respectively.
Prove that while merging H1 and H2, the number of times two smaller trees are merged into a bigger tree, is equal to the number of carry-overs generated when B1 and B2 are added.