For a given random sequence (C,T1,T2,âŠ), the smoothing transform S maps the law of a real random variable X to the law of âkâ„1TkXk+C, where X1,X2,⊠are independent copies of X and also independent of (C,T1,T2,âŠ). This law is a fixed point of S if X=dâkâ„1TkXk+C holds true, where =d denotes equality in law. Under suitable conditions including EC=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T1,T2,âŠ) such that the canonical solution exhibits right and/or left Poissonian 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.