How about one of those self-charging emergency flashlights that you shake to charge? Fasten the light vertically to the base of the tree, drill a hole, or disassemble the flashlight so you can attach a string to the little sliding magnet part inside the light. Run the string up to a pulley, then over to the branch you want to measure. Remove the light bulb and attach your voltmeter and microamp meter to read the power output. That may only give instantaneous measurements, so if you wanted to measure over time, measure the built-in battery or supercap's beginning (discharged) voltage, and after a time, measure again. some basic math for calculating voltage difference over amount of time would tell you how much power could be achieved from that specific branch.  I'm sorry, my memory is really fuzzy right now regarding said math formula...should be easy enough to find though..  There are several energy projects going on in recent years which involved harvesting energy from waves and tides, one of which is on the sourthern Oregon coast. Perhaps you could get some more ideas from reading about those! 

piezo type setup

Hi, sorry, I posted before seeing the rest of the post here. I think this area (again I call it micro power harvesting), is interesting and their are many ways, and one of the trickier parts is picking the right 'generator' to match what driving it. 

A few things I found that narrow down which gen fits are things like the frequency (in the backpack pendulum thing it would be like ~1 hz, in the piezo "humdinger" it is 10'S of KZ (I think).

But I do agree, agp.cooper, that the stepper motor (both linear and conventional 'round' like in any electric bike now a days, is just waiting to be utilized more and more.

On the motor as a generator, it is tempting to find the 'strong' motor (the ones where you can feel the torque even when turning with your hands, but you must have the drive force to overcome this. Can happen, even if this was in a shoe, with a strong linear motor like a shock, but the weight of your body does the compression, and charges some battery. The use of the alternating line of strong magnets/coils is something I think can really be leveraged in many ways, both for new motor/moving things as well as in reverse for novel power generation in awkward to recharge/remember to recharge etc. applications

Actually, I think this maybe have some legs to it (no pun intended). It so far has been mostly thought experiment, but I did play with it a bit.

Here is one scenario that it could have a place. I have been interested in what some call 'micro-power harvesting', it is important to keep in mind the micro, it is small, but for some low power devices, this could be a way that you don;t have to keep charging those little batteries (c'mon, we all forget to charge, especially when the number of gadets we start to carry adds up, and if they are maybe buried inside some wearable (my idea at that time was battery pack, sewed in (not loosely bouncing aorund), and well a pendulum setup for charging as you walk etc. In this regard the 'pendelum' mostly organizes (constrains) the motion to match up with your power generator (if you use a traditional coil/magnet system). You can do other ways (piezo maybe, but actually the quite low frequency (and size weight/space) in a backpack example lends itself (maybe) to fitting this. 

This idea in principle is "proven', but he old style mechanical 'self winding' watches (of decades ago... lol  .  So while the pendelum doesn't give you power, a key part is you DO WANT a "high Q" (low frictional loss system, as you are trying to capture and store what micropower you can grab.  

For things blowing in the wind there was a device (company-  Humdinger I think,  

http://www.physics.org/featuredetail.asp?id=47

I also been thinking of something similar to replace the batteries in wireless mouse (funny, for that one the best idea I had was go back to utilizing the old "ball mouse" but use the ball for the generator (again micro harvest, but probably enough to charge the battery)..

just sharing, 

I don't think you want a pendulum.

Attach a spring to the branch and measure the amount the spring is extended as the branch sways, then the energy is the potential energy of the spring:

http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html

A tiny pendulum can't store much energy. Model it, and you'll find out how much energy you can extract from the pendulum before damping it too much.

I would use a stepper motor (just being practical) and a schottky diode bridge.

AlanX