Are all trees in Fibonacci heaps binomial trees?

Question

Is it possible for a Fibonacci heap to contain a tree that isn't a binomial tree? If so, how would this happen? Can you give an example?

Answer 1

Yes, this can happen. Intuitively, the reason is that in a Fibonacci heap, the decrease-key operation can work by cutting a subtree from a larger tree, resulting in two trees that are (potentially) not binomial trees. This differs from the binomial heap, where decrease-key works by doing a bubble-up operation from the node whose key was decreased all the way up to the root.

To see a concrete example, let's insert five elements into a Fibonacci heap, say, 1, 3, 5, 7, and 9. This gives the heap

1 - 3 - 5 - 7 - 9

Now, let's do a dequeue-min, which extracts 1. We now try to compact all of the remaining elements together, which merges the trees as follows:

    3
   /|
  5 7
    |
    9

Now, suppose that we do a decrease-key operation on to decrease the key of 9 to 6. To do this, we cut 9 from its parent and merge it into the list of trees at the top, which yields

   3 - 6
  /|
 5 7

And now the tree with 3 at its root contains only 3 elements, so it is not a binomial tree anymore.

Hope this helps!