On the growth of components with non fixed excesses

Denote by an $l$-component a connected graph with $l$ edges more than vertices. We prove that the expected number of creations of $(l+1)$-component, by means of adding a new edge to an $l$-component in a randomly growing graph with $n$ vertices, tends to 1 as $l,n$ tends to $\mathrm{\infty }$ but with $l = o(n^{1/4})$\mathrm{.}Wealsoshow,underthesameconditionson$l$and$n$, that the expected number of vertices that ever belong to an $l$-component is $\sim (12l)^{1/3} n^{2/3}$.