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The number of octrees with $n$ nodes is OEIS sequence A007556: $$T(n) = {8n \choose n}/(7n+1)$$ I worked this out, incidentally, by writing a short program to generate the first few values of $T(n)$ and then searching for it. If you had no other information than $n$, then the minimum number of bits required to store an octree with $n$ nodes is $\log\,T(n)$...


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