  We compute the Newton--Okounkov bodies of line bundles on a Bott--Samelson
resolution of the complete flag variety of $GL_n$ for a geometric valuation
coming from a flag of translated Schubert subvarieties. The Bott--Samelson
resolution corresponds to the decomposition
$(s_1)(s_2s_1)(s_3s_2s_1)(\ldots)(s_{n-1}\ldots s_1)$ of the longest element in
the Weyl group, and the Schubert subvarieties correspond to the terminal
subwords in this decomposition. We prove that the resulting Newton--Okounkov
polytopes for semiample line bundles satisfy the additivity property with
respect to the Minkowski sum. In particular, they are Minkowski sums of
Newton--Okounkov polytopes of line bundles on the complete flag varieties for
$GL_2$,\ldots, $GL_{n}$.
