An essential feature of Newton polytopes of polarized toric varieties is the additivity property with respect to the Minkowski sum, that is, tensor product of line bundles corresponds to the Minkowski sum of Newton polytopes. In particular, this property is crucial for the famous Bernstein-Koushnirenko theorem. The additivity property does not necessarily hold for Newton-Okounkov convex bodies of more general varieties and valuations. We show that the additivity property holds for a geometric valuation on a Bott-Samelson resolution of the variety of complete flags in C$^n$. The resulting Newton-Okounkov polytopes are combinatorially different from previously known polytopes, and can be obtained as Minkowski sums of Newton-Okounkov polytopes of varieties of complete flags in C$^2$ ,…, C$^n$ .