of the global leaf removal process. For instance, $p_0$ counts rooted trees with an isolated root, hence trees with a single vertex, and $p_0=e^{-\alpha}$. As another example, $q_1$ counts rooted trees whose root touches at least one standard leaf. Consider the trees whose root touches exactly $k\geq 1$ standard leaves, and $l\geq 0$ other vertices. These $l$ vertices can be seen as roots of non-trivial subtrees of the original tree, so by definition they contribute to $1-p_0$. So the weight is $$e^{-\alpha} \frac{(\alpha e^{-\alpha})^k}{k!}\frac{(\alpha
  (1-p_0))^l}{l!}.$$ Hence $$q_1=e^{-\alpha} \sum_{k \geq 1} \sum_{l \geq 0} 
\frac{(\alpha e^{-\alpha})^k}{k!}\frac{(\alpha
  (1-p_0))^l}{l!}=1-e^{-\alpha e^{-\alpha}}.$$

In the same way, contributions to $p_n$ or $q_n$ for larger $n'$s can be analyzed in terms of the trees attached to the neighbors of the root, and the structure of these trees involves lower order contributions. In contributions to $p_n$, the root has at least one neighbor whose attached tree contributes to $q_n$ and any number of neighbors contributing to $q_1$ or $q_2$ or $\cdots$ or $q_{n-1}$. So $$p_n=e^{-\alpha}(e^{\alpha q_n}-1)e^{\alpha (q_{n-1}+\cdots+q_1)}.$$ Analogously, in contributions to $q_n$, the root has at least one neighbor whose attached tree contributes to $p_{n-1}$ and any number of neighbors none of which contributing to $p_0$ or $p_1$ 