  This paper studies the distribution of the component spectrum of
combinatorial structures such as uniform random forests, in which the classical
generating function for the numbers of (irreducible) elements of the different
sizes converges at the radius of convergence; here, this property is expressed
in terms of the expectations of {\it independent} random variables $Z_j$,
$j\ge1$, whose joint distribution, conditional on the event that $\sum_{j=1}^n
jZ_j = n$, gives the distribution of the component spectrum for a random
structure of size $n$. For a large class of such structures, we show that the
component spectrum is asymptotically composed of $Z_j$ components of size $j$,
$j\ge1$, with the remaining part, of size $n-\sum_{j\ge1} Z_j$, being made up
of a single, giant component.
