For the already assembled Christmas tree, I want to be able to follow the circuit and say "this is the reason for ____ and it is connected to ____ because of ..." In other words, I want to be able to do more than just solder parts together.
I agree, you don't learn much from blinding soldiering components together, children working in sweetshops do that all day and know less about electronics than you.
Have a look on Wikipedia for a good article on astable multivibrators.
http://en.wikipedia.org/wiki/Astable_multivibratorI want to understand the why behind it. Moving into micro-controllers feels like a natural progression, to me at least, when it comes to making something do that off the shelf parts (requires no programming) are unable to do. It opens up more possibilities than saying "I don't know how to do that, I guess I better find a different solution."
Moving on to microcontrollers is a good idea but it's a good idea to make sure you understand the fundamentals first.
Whilst microcontrollers are the most flexible parts there are lots of very simple problems which are actually better solved with discrete components.
I would suggest learning a bit about op-amps, comparators and logic gates as well as microcontrollers.
Adding more LEDs is a purely cosmetic thing to improve the looks of it. The reason I want to stick with the 9v battery is because right now I have some double sided tape on the bottom of the battery and the back of the circuit board. When you look at the "tree" straight on, it looks like it is standing Once I get all the details worked out, a prototype working correctly, I want to put together a small handful of them and hand them out as little gifts - another reason why I am sticking to a standard 9v battery over coin batteries for instance.
I noticed the circuit contains a couple of drawing errors: the bottom cell is upside down, dots aren't always used to connect wires and standard convention is to have the battery on the left with the switch on the positive.
Here's how to add more transistors to increase the power output so you can drive more LEDs. The extra PNP transistors enable it to switch more LEDs connected in parallel. Note also how I've only put three in series rather than four so it will work at a lower voltage.