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Amirmohammad Farzaneh, Mihai-Alin Badiu, and Justin P. Coon, <a href="https://arxiv.org/abs/2309.09779">On Random Tree Structures, Their Entropy, and Compression</a>, arXiv:2309.09779 [cs.IT], 2023.
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Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RootedTree.html">RootedTreeRooted Tree</a>
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digits = 87; max = 250; s[n_, k_] := s[n, k] = a[n+1-k] + If[n < 2*k, 0, s[n-k, k]]; a[1] = 1; a[n_] := a[n] = Sum[a[k]*s[n-1, k]*k, {k, 1, n-1}]/(n-1); A[x_] := Sum[a[k]*x^k, {k, 0, max}]; APrime[x_] := Sum[k*a[k]*x^(k-1), {k, 0, max}]; eq = Log[c] == 1 + Sum[A[c^(-k)]/k, {k, 2, max}]; alpha = c /. FindRoot[eq, {c, 3}, WorkingPrecision -> digits+5]; b = Sqrt[(1 + Sum[APrime[alpha^-k]/alpha^k, {k, 2, max}])/(2*Pi)]; RealDigits[b, 10, digits] // First (* _Jean-François Alcover_, Sep 24 2014 *)
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