Please change the title if it's not accurate, I might be lacking terminology.
I have two variables ($x,y$) which are directly correlated by pair. For example I have sampled 50 trees in my study area and for each tree I have tree height and annual precipitation at that location. This is what I mean by they are organized by pair. So I plot $x$ vs $y$ and I would like to know if there is any correlation between the two; does annual precipitation predict tree height, or the other way around?
I have used R to test if there is any linear correlation with the function
lm(y~x) and have $p<.05$ and $R^2=0.6$, but some of my data seem as if they might be better fitted with a non-linear curve. What are some other types of statistical analysis I should use for such a problem?