Reverse engineering is basically analyzing the system to figure out how it works, then coming up with a something similar without ever seeing the original design documentation. I saw one of those trees at home depot, but thought the price was a little high. They went pretty fast though...
If you are not already well versed in electronics and have some computer programming ability, you are in for an uphill climb.
I would start by looking at how the thing is wired and watch the programming very carefully - that will help you to determine whether the lights are controlled directly or through a network of some sort.
If it is direct controlled, you will have to look at the number of channels and lights per channel and see if something like a MR16 could be used. If it is direct controlled, there will be a lot of wires coming from the control box.
If it is some sort of network, you need to figure out the communication protocol - either that or just replace all of the lights with smart strings... Assuming that it is a network, it may use one of the common chips, which would make things easy. If it is a 3 or 4 wire controller, You could just hack off the controller and substitute a new one. The E681 controller does several formats and also supports both 5 and 12 volt strings.
If it is a 3 or 4 wire interface and does not use one of the common chip types, you will need either an oscilloscope or a logic analyzer to look at the data stream coming out of the controller and try to analyze the data relative to what the tree is doing. One gotcha to look out for is string initialization. most of the strings do not require initialization, but the GE35 strings require each node to be given an address by the controller before they will operate.
Good luck!
RM