  We show that on a metric graph of genus $g$, a divisor of degree $n$
generically has $g(n-g+1)$ Weierstrass points. For a sequence of generic
divisors on a metric graph whose degrees grow to infinity, we show that the
associated Weierstrass points become distributed according to the Zhang
canonical measure. In other words, the limiting distribution is determined by
effective resistances on the metric graph. This distribution result has an
analogue for complex algebraic curves, due to Neeman, and for curves over
non-Archimedean fields, due to Amini.
