= Natural-sum ordinal rank of a finite rooted tree
{title2=$o(T)=\mathop{\#}_{U\text{ child subtree}}\omega^{o(U)}$}
Give a leaf rank zero, and use <Hessenberg natural sum> over the immediate rooted subtrees. Every rank is below <epsilon zero>; conversely repeated exponentiation and finite summation encode every <Cantor normal form> below it. A <rooted-tree homeomorphic embedding> implies nondecrease of the rank. When roots match, child subtrees embed into distinct branches and exponentiation and natural sum preserve their inequalities. When the source root maps below the host root, its rank is bounded by that of a proper subtree, strictly below the host rank.
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