  In this paper, we study tropical Weierstrass points. These are the analogues
for tropical curves of ramification points of line bundles on algebraic curves.
  For a divisor on a tropical curve, we associate intrinsic weights to the
connected components of the locus of tropical Weierstrass points. This is
obtained by analyzing the slopes of rational functions in the complete linear
series of the divisor. We prove that for a divisor $D$ of degree $d$ and rank
$r$ on a genus $g$ tropical curve, the sum of weights is equal to $d - r + rg$.
We establish analogous statements for tropical linear series.
  In the case $D$ comes from the tropicalization of a divisor, these weights
control the number of Weierstrass points which are tropicalized to each
component.
  Our results provide answers to open questions originating from the work of
Baker on specialization of divisors from curves to graphs.
  We conclude with multiple examples which illustrate interesting features
appearing in the study of tropical Weierstrass points, and raise several open
questions.
