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Binomial Heap Properties

1.: Each binomial tree is heap-ordered (key(x) $\geq$ key(parent(x)).
This is the opposite of previous heap properties.
2.: There never exist two or more trees with the same degree in the heap.

A binomial heap with n nodes has at most $\lfloor \lg n \rfloor \;+\; 1$ binomial trees.

n in binary = <b_k, b_k-1, .., b₀> bits
k = $\lfloor \lg n \rfloor$ , n = $\sum_{i=0}^{\lfloor \lg n \rfloor} b_i 2^i$

There is a one-to-one mapping between the binary representation and binomial trees in a binomial heap.

: If b_i = 1, then B_i is in the heap
: Recall that there are 2ⁱ nodes in B_i

At most $\lfloor \lg n \rfloor$ + 1 bits are needed to express n base 2

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