When I was studying the structure of the Grigorchuk group, a question came to my mind and I just had the following information:

We know that every finitely generated group of subexponential growth is amenable so its Traski number is infinity. Also if a group $G$ contains free group on two generators then we know from Jonsson-Dekker theorm that its Tarski number is 4. And from another theorem we can conclude that $G$ has exponential growth.

But I couldn't understand that, is there any relation between the Tarski number of a group and its growth, or not? 

If I want to say what I mean, Let a group $G$ have Tarski number $k$, Is the growth function $\gamma_G (n)$ dependent to $k$ or not? Also if the growth function $\gamma_G (n)=\lambda$, is the Tarski number $\tau (G)$ dependent on $\lambda$ or not?