process at step $n$ (which goes from $G^{(n-1)}$ to $G^{(n)}$): emergence of a new isolated point, removal of a point, removal of an edge[^1]. This enumeration is possible because the finite configurations of vertices and edges in the random graph with extensive multiplicity are tree-like. This implies that the problem has a recursive structure. The weight of a tree is chosen so as to reproduce the correct random graph weight: a vertex has weight $e^{-\alpha}$ and an edge has weight $\alpha$. One has to be rather careful to avoid double counting and omissions; the examination of all cases is very tedious so we omit the details and simply outline the strategy.

The key ingredient is a study of leaf removal on rooted trees. Starting from a rooted tree, we apply the global leaf removal process, with the convention that even if the root has only one neighbor, it is not counted as a standard leaf[^2]. We let $p_n,\; n\geq 0$ be the generating function for rooted trees whose root becomes isolated exactly at step $n$ of the global leaf removal process, and $q_n,\; n\geq 1$ be the generating function for rooted trees whose root is removed exactly at step $n$ 