  For a given random sequence $(C,T_{1},T_{2},\ldots)$ with nonzero $C$ and
a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform
$\mathcal{S}$ maps the law of a real random variable $X$ to the law of
$\sum_{k\ge 1}T_{k}X_{k}+C$, where $X_{1},X_{2},\ldots$ are independent copies
of $X$ and also independent of $(C,T_{1},T_{2},\ldots)$. This law is a fixed
point of $\mathcal{S}$ if the stochastic fixed-point equation (SFPE)
$X\stackrel{d}{=}\sum_{k\ge 1}T_{k}X_{k}+C$ holds true, where $\stackrel{d}{=}$
denotes equality in law. Under suitable conditions including $\mathbb{E} C=0$,
$\mathcal{S}$ possesses a unique fixed point within the class of centered
distributions, called the canonical solution to the above SFPE because it can
be obtained as a certain martingale limit in an associated weighted branching
model. The present work provides conditions on $(C,T_{1},T_{2},\ldots)$ such
that the canonical solution exhibits right and/or left Poisson tails and the
abscissa of convergence of its moment generating function can be determined. As
a particular application, the right tail behavior of the Quicksort distribution
is found.
