Is this a legal Fibonacci heap? Imagine a Fibonacci heap with 1 tree: a root node, 4 child nodes (to that root node), with 2 of them being leaves and the other 2 having 1 child each (7 nodes total). Is this a legal Fibonacci heap? In other words, does there exist a series of actions that lead to this sort of heap structure.


My intuition tells me no, because it just seems too "unstructured". On the other hand, it doesn't seem to violate the max degree property or any other basic properties of a Fibonacci heap.

I thought about extracting the minimum and seeing if that'll lead to an illegal heap, but it doesn't seem to be the case.
It's not possible. You propose a root node with degree 4 and size 7 (including itself).

From CLRS 3ed pp 525:


  Lemma 19.4
  
  Let $x$ be any node in a Fibonacci heap, and let $k = x.degree$. Then $size(x) \geq F_{k + 2}$.


$$
\begin{align}
k &amp; = 4\\
F_{6} &amp; = 8\\
7 &amp; \ngeq 8\\
\end{align}
$$