# Interpretation of the coefficients in the sum of positive roots

Take a finite Cartan datum with index set $$I$$, simple roots $$\{\alpha_i\mid i \in I\}$$ and positive roots $$\Phi^+$$. Let $$2\rho=\sum_{\alpha\in\Phi^+}\alpha$$ be the sum of the positive roots and write $$2\rho = \sum_{i\in I}n_i\alpha_i.$$You can find tables of these coefficients, for example, in the appendix to Bourbaki, chapters 4-6.

My question: what is the significance of the numbers $$n_i$$? In particular, do they count anything?