or $\cdots$ or $p_{n-1}$. So $$q_n=e^{-\alpha}(e^{\alpha p_{n-1}}-1)e^{\alpha
  (1-p_{n-1}-\cdots-p_0)}.$$ These two relations allow it to be shown that $$\begin{aligned}
 p_n & = & e_{2n+1}-e_{2n-1} \quad \mathrm{for} \; n
\geq 0 \\  q_n & = & e_{2n-2}-e_{2n} \quad \mathrm{for} \;
 n \geq 1, \end{aligned}$$ where $e_n(\alpha)$ is the sequence of iterated exponentials, defined by $$e_{-1}=0 \quad \mathrm{and} \quad e_n=e^{-\alpha e_{n-1}} \quad 
 \mathrm{for} \; n \geq 0.
 \label{eq:en}$$

The events on the random graph that at step $n$ of the global leaf removal process a given vertex becomes isolated, or that a given vertex disappears, or that a given edge disappears can all be interpreted in terms of the previous configurations. In each case, the different possible contributions have to be taken into account, and also the rule that a root with a single neighbor can be touched by leaf removal has to be enforced. We omit this painful case by case analysis and only state the results.

The explicit formulæ for extensive contribution to the average number $Ni_n(\alpha)$ of isolated vertices, $Nc_n(\alpha)$ of non isolated vertices and $Nl_n(\alpha)$ of edges in $G^{(n)}$ are $$\begin{aligned}
  \label{eq:incndn}
  i_n(\alpha) & = & e_{2n+1}+e_{2n}+\alpha e_{2n}e_{2n-1}-1, \nonumber \\
  c_n(\alpha) & = & e_{2n}-e_{2n+1}-\alpha e_{2n}e_{2n-1}+\alpha
  e_{2n-1}^2,  \\ 
  l_n(\alpha) 