There were 4 solutions received. Kevin's was the only one that was both correct and complete.

Now to the answer:

Let x denote the vertical distance from where the wire is attached to the trees to the point where the descending wire begins. Then x lies in the interval [0,d] with x = 0 corresponding to the "T"--shaped configuration and x = d corresponding to the "V"-shaped configuration.

The amount of wire is easily seen to be

Our task is to minimize this quantity over the interval [0,d]. Straightforward differentiation gives a unique local minimum at the point

This local minimum is a global minimum as long as it is a point in the interval in question, i.e. as long as P < d. In the other case the "V"-shaped configuration yields a minimum.

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