  Let $f$ be a function in the Eremenko-Lyubich class $\mathcal{B}$, and let
$U$ be an unbounded, forward invariant Fatou component of $f$. We relate the
number of singularities of an inner function associated to $f|_U$ with the
number of tracts of $f$. In particular, we show that if $f$ lies in either of
two large classes of functions in $\mathcal{B}$, and also has finitely many
tracts, then the number of singularities of an associated inner function is at
most equal to the number of tracts of $f$.
  Our results imply that for hyperbolic functions of finite order there is an
upper bound -- related to the order -- on the number of singularities of an
associated inner function.
