  Let f be an entire function that has only finitely many critical and
asymptotic values. Up to topological equivalence, the function $f$ is
determined by combinatorial information, more precisely by an infinite graph
known as a "line-complex". In this note, we discuss the natural question
whether the order of growth of an entire function is determined by this
combinatorial information.
  The search for conditions that imply a positive answer to this question leads
us to the "area property", which turns out to be related to many interesting
and important questions in conformal dynamics and function theory. These
include a conjecture of Eremenko and Lyubich, the measurable dynamics of entire
functions, and pushforwards of quadratic differentials.
  We also discuss evidence that invariance of order and the area property fail
in general.
