  The main result of this note is that the toric degenerations of flag
varieties associated to string polytopes and certain Bott-Samelson resolutions
of flag varieties fit into a commutative diagram which gives a resolution of
singularities of singular toric varieties corresponding to string polytopes.
Our main tool is a result of Anderson which shows that the toric degenerations
arising from Newton-Okounkov bodies are functorial in an appropriate sense. We
also use results of Fujita which show that Newton-Okounkov bodies of
Bott-Samelson varieties with respect to a certain valuation $\nu_{max}$
coincide with generalized string polytopes, as well as previous results by the
authors which explicitly describe the Newton-Okounkov bodies of Bott-Samelson
varieties with respect to a different valuation $\nu_{min}$ in terms of
Grossberg-Karshon twisted cubes. A key step in our argument is that, under a
technical condition, these Newton-Okounkov bodies coincide.
