The rational index ρ_L of a language L is an integer function, where ρ_L(n) is the maximum length of the shortest string in L∩ R, over all regular languages R recognized by n-state nondeterministic finite automata (NFA). This paper investigates the rational index of languages defined by (context-free) grammars with bounded tree dimension, and shows that it is of polynomial in n. More precisely, it is proved that for a grammar with tree dimension bounded by d, its rational index is O(n^{2 d}), and that this estimation is asymptotically tight, as there exists a grammar with rational index Θ(n^{2 d}).