  We consider paths of steepest descent, in the complex plane, for the norm of
a non-constant one variable polynomial $f$. We show that such paths, starting
from a zero of the logarithmic derivative of $f$ and ending in a root of $f$,
draw a tree in the complex plane, and we give an upper bound estimate on their
lengths. In some cases, we obtain a finer estimate that depends only on the set
of roots of $f$, not on their multiplicity, and we wonder if this can be done
in general. We also extend this question to finite Blaschke products for the
unit disk.
