  We study randomly growing trees governed by the affine preferential
attachment rule. Starting with a seed tree $S$, vertices are attached one by
one, each linked by an edge to a random vertex of the current tree, chosen with
a probability proportional to an affine function of its degree. This yields a
one-parameter family of preferential attachment trees $(T_n^S)_{n\geq|S|}$, of
which the linear model is a particular case. Depending on the choice of the
parameter, the power-laws governing the degrees in $T_n^S$ have different
exponents.
  We study the problem of the asymptotic influence of the seed $S$ on the law
of $T_n^S$. We show that, for any two distinct seeds $S$ and $S'$, the laws of
$T_n^S$ and $T_n^{S'}$ remain at uniformly positive total-variation distance as
$n$ increases.
  This is a continuation of Curien et al. (2015), which in turn was inspired by
a conjecture of Bubeck et al. (2015). The technique developed here is more
robust than previous ones and is likely to help in the study of more general
attachment mechanisms.
