A binomial forest is a very interesting data structure. But, to discuss it, we need to first start with a binomial tree. A binomial tree is a tree in which a combination of two smaller binomial trees of the same size are combined in a particular way:

Binomial tree

The preceding figure shows how binomial trees combine to create larger binomial trees. In the first row, two binomial trees of height 1 combine to create a new binomial tree of height 2. In the second row, two binomial trees of height 2 combine to create a new binomial tree of height 3. In the final example, two binomial trees of height 3 combine to create a binomial tree of height 4, and it continues. The two trees that are combined together are not treated symmetrically. Instead, the root of one becomes the parent of the other. The next figure shows one more step in the sequence and then shows a different way to look at a binomial tree. In the last row, I have highlighted the subtrees differently. Notice how:

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