Rooted-tree generating function ($T(z)={\sum }_{j\ge 1}{j}^{j-1}{z}^j/j!$)

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 $0\le z<1/e$, 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.