Is there any kind of electrical structure inside the tree that generates a voltage differential between the roots and the branches? I've never heard of one.
I have seen a few mV differential between my left hand holding one lead, and my right hand holding the other lead. I cannot account for this, unless it might be static charge, but that isn't satisfactory either.
There are reports of electrical signals observed coming from a probe (or probes) attached to plants.
If the electrodes are of dissimilar metals, then there may be some current generated. The method is similar to the 'potato powered clock' available as a commercial kit.
I believe a tree could pick up magnetic 60 cycle hum in nearby power lines. This would show up on an oscilloscope.
There are believable reports of getting power (in small amounts) by hanging a wire a few stories aboveground. If you were to test a tree, you would need to insulate the probes, in order to know it wasn't due to electricity from the air.