= Solution
Let $\beta$ be a positive root. If it is not simple, it is a sum of two positive roots. Repeating this decomposition terminates because the height with respect to a regular positive functional strictly decreases, and it writes
$$
\beta=\sum_i n_i\alpha_i,
\qquad n_i\in\mathbb Z_{\geq0},
$$
with at least one $n_i>0$. Equivalently, one may repeatedly choose a simple root $\alpha_i$ with $(\beta,\alpha_i)>0$ and use the <root string> to replace $\beta$ by the root $\beta-\alpha_i$.
For uniqueness, the simple roots are linearly independent. Therefore two such expansions have identical coefficients. Here “positive integer coefficients” must allow zero coefficients: a simple root itself has coefficient one on its own basis vector and zero on the others.
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