Now don't get me going, you know I like this sort of thing.

The sap rises in the Spring because it gets warmer, and the tree's metabolism picks up. Warmer means more power from the Sun hitting the bit of the Earth where the tree is. Now, we know pretty well the power that hits the Earth from the Sun: it's about 1360 kW (kilowatts) per square kilometre, and it goes up and down with the seasons because of the tilt of the Earth's axis.

Because the Moon is in about the same orbit as the Earth from the Sun's point of view, we can take that as the power figure reaching the Moon too. The Moon has a radius of about 1738km, it has an area (thinking of it as a flat disc) of 9.5 million square kilometres. So the total power hitting the Moon is 9.5e6(*) x 1360 = 13000 million watts (1.3e10W). The Moon has an "albedo"- reflectivity- of 0.136, so it re- radiates a power of 1.3e10 x .136 = 1.8e9W.

The radius of the Moons orbit is about 385000km. Think of the moon re- radiating its power over a hemisphere the size of its orbit, towards the Earth (we're not interested if the power goes the other way, that's not full moon). 2 x Pi x radius squared: that's the surface of a hemisphere. 2 x Pi x 385000 squared = 0.9 billion square kilometres (9e11). So at the Earth, the Moon is sending us just 1.8e9/9e11 = 2 milliwatts (thousandths of a watt) per square kilometre.

That's not a lot: two milliwatts is a lot less than what your mobile phone takes on standby (it takes a few watts when you're using it).

Even a big tree is nothing like a square kilometre in area, so I think we have to look for another possible source of influence if there IS any effect on trees. I'll do gravity when you've all woken up.

As for it's effect on people, that we can see by moonlight is a tribute to the wonderful nature of our eyes, that can adapt to light levels over a power range of about a billion to one (the range from bright sunlight to quarter- moon is only a few million to one). Our eyes have evolved to cope with the range of useful illumination available.

(*) This 'e' means "This number is followed by the following number of zeros", and is commonly used in computer programming and engineering to save us writing lots of zeros. Mathematicians don't like it because they've got a different 'e' which is a fixed number a bit like Pi, but with a value of about 2.72.