  The performance of low-density parity-check (LDPC) codes in the error floor
region is closely related to some combinatorial structures of the code's Tanner
graph, collectively referred to as {\it trapping sets (TSs)}. In this paper, we
study the asymptotic average number of different types of trapping sets such as
{\em elementary TSs (ETS)}, {\em leafless ETSs (LETS)}, {\em absorbing sets
(ABS)}, {\em elementary ABSs (EABS)}, and {\em stopping sets (SS)}, in random
variable-regular and irregular LDPC code ensembles. We demonstrate that,
regardless of the type of the TS, as the code's length tends to infinity, the
average number of a given structure tends to infinity, to a positive constant,
or to zero, if the structure contains no cycle, only one cycle, or more than
one cycle, respectively. For the case where the structure contains a single
cycle, we obtain an estimate of the expected number of the structure through
the available approximations for the average number of its constituent cycle.
These estimates, which are independent of the block length and only depend on
the code's degree distributions, are shown to be accurate even for
finite-length codes.
