Choosing response and predictors 
Information on the Girth (), Height () and Volume () for 31 cherry trees
yielded the following correlations : $r_{12}$ = 0.519,  $r_{23}$= 0.598 and  $r_{13}$= 0.967. If you were asked to setup a regression choosing a certain characteristic as the response
and the other two as predictors, which characteristic will you choose as the
response? Give your comments.

I think we choose Girth and Height as predictors for the response Volume because the correlation between Girth and Height is the least, so we're looking at how these individually affect the Volume (i.e. the factor with which their correlation is more than that between them).
Is my reasoning correct?
This question was asked in the final examinations of the course on Multivariate Data & Linear Models and it had 10 marks. What more should I write?
 A: I might argue that it makes sense for volume to be the response, and then you can use direct measurements of how tall and wide a tree is in order to predict the volume (which is probably harder to measure).
However, we use regressions to predict or explain. If we want to predict or explain differences in girth, we would use girth as the response. If we want to predict or explain height, we would use height as the response. If we want to predict or explain volume, we would use volume as the response.
In all three cases, the other two variables are viable predictors.
I do not see the correlations between the marginal variables playing any role. If I wanted to measure volume and girth ($r=0.967$) in order to predict height, I would be totally comfortable doing so. One might argue that it might be worth it to drop one of those correlated predictors in order to have fewer parameters without losing much in terms of predictors (since the volume and girth have such a high correlation). While I understand this stance, it is not so simple.
A: Predictors with high correlation are more linearly dependent. So they have a similar effect on the dependent variable. Thus doesn't make sense to use Height as the response. Choosing between Girth and Volume it seems more logical to use Volume. As you mentioned in the answer to @whuber, the volume might be the most difficult to measure in living trees.
A: The data alone does not suggest a model, just a correlation matrix.  What is the purpose of the model, and in what conditions will it be applied?  For example, if the trees are still standing in a grove or forest, it might be useful to estimate their height, based on girth (usually measured at 48" above ground).  Of course, this is an opportunistic dataset collected from a plot of felled cherry trees, so the lumber mill model, volume ~ girth + height may be the one you want.
