Holy Moly that's a project.
1. LED's are all wired together. No individual control.
2. LED's are only white. Think I've seen some trees where the color changes.
3. Tree plug directly into 110 volts. Indicating the LEDs are wired in series each with it's own resistor. So they are like Christmas light and do not need a power supply.
You're not going to like this, but I would recommend you practically start over. Get a "blank" tree and some of these LED strings:https://www.adafruit.com/products/322
You don't need anything as powerful as a RaspberryPi. You can control these using an Atmel based controller like an Arduino. And they can be programmed to display what ever color you wish.
I can see you are not moved. Ok, let's see what you have got your self into. You want to control 600 white LEDs. They are probably super bright so can not be powered directly by any processor. Here, let's use these for an example:https://www.adafruit.com/products/754
These (which may or may not be like yours) run on 3V at about 20mA. Ok, so to light up 600 of these you need a power supply that is 3V at 12 amps. To be safe we might want to double that. So 3V at 24 amps. Or 72 watts. Hum.
It is likely that in any design you use, you will use a resistor in series with the LED. In such a case you can use a higher voltage power supply and drop the difference across the resistor. Like use a 5V supply and drop 2V across the resistor and the other 3V across the LED. If the expected current is .02A then, to drop 2V, you can try using a 100ohm resistor.
Most people are going to use a MOSFET transistor to switch on and off a high current LED. Here is a web page that talks about controlling LEDs for a Stop Sign Costume:http://systembus.com/110621/%5BnRF24%5Ddiyembedded2/www.sparkfun.com/tutorials/205.htm
Half way down the page you see a schematic which includes 2 resistors and a FET. He is controlling 9 LEDs that draw about 400mA all together. You want to control each LED individually. So you need to adjust the resistor in series with your LED according to your power supply and the current your LEDs will draw. So, for every LED you have, you need to buy 2 resistors and a FET and wire this all in someplace on the tree. Then you need to bring back 600 wires to the RaspberryPi. Which, BTW, does not have 600 GPIO pins.
That's too much work. What else is there?
Well, there is what you suggested. Using the Maxim IC. A quick look says you should be able to control 18 LEDs? So that puts you at 4 chips. You could place them around the tree and wire back the LEDs to the chips.
Note, this is something very similar to that string of LED in the first link above.
Are there other ways of doing this? Yes, you can try building a matrix and wiring up each of the 600 LEDs to coordinates in that matrix. That way you only need two times the square root of the number of LED worth of switching circuits. It's a bit tricky to design using FETs, but it should be possible. So a matrix of 30 by 20 should give you control over 600 LEDs. Of course you can only light up one LED at a time. Not so different from the way a TV screen is scanned. But you realize to get a frame rate of only 30 FPS you would have to plow through your matrix at a rate of 18000 LEDs a second.
Are there other ways? I can think of a few more bizarre ones. But I think the most sane is to go with the 1st link above.