= Rooted-tree generating function
{title2=$T(z)=\sum_{j\geq1}j^{j-1}z^j/j!$}
The <exponential generating function> for labelled <rooted trees> satisfies $T=z e^T$: remove the root and obtain an unordered set of smaller labelled <rooted trees>. For $0\leq z<1/e$, the convergent series is the solution in $[0,1)$ of $T e^{-T}=z$, rather than the larger real solution. Alternatively, <tree-component expectation in the Erdős-Rényi model> and subcritical exploration give $T(\theta e^{-\theta})=\theta$ for $0<\theta<1$: their limiting probabilities for the order of the <tree component> of a uniform <vertex> sum to one. The component tail bound makes this passage through the infinite sum valid.
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