Concerns About Solar Charging Tree House Lights
There are a number of issues here. In no particular order:


The solar panels should each have their own series blocking diode. If one solar panel is covered and getting partial shade, its output will be less than the other panels and will be a power sink instead of a power source. 
The solar panels deliver 4.5V at 80mA at full sun. Under partial sun they won't deliver enough voltage to charge the batteries. 
The rechargeable batteries have a voltage of 1.2V, not 1.5V. This means your battery bank has a 3.6V output, not a 4.5V. This works to your advantage as the LEDs will still work (typically from 3V - 5V), and the solar panels minus the diode voltage drop is still enough to charge the batteries under full sun.  
Rechargeable batteries like this need to charge at a rate of 10% of the aH rating for 16 hours, for a total of 160% energy in to get 100% energy out. You may need to let this charge for 2 full days to get the batteries to full charge, under full sun.


Given that, will the system charge and discharge correctly? Under full sun, yes, your system will work as expected. The light output may be a little lower than if you were running off of non-rechargeable batteries, but not enough to notice.

Will the lights be powered with the correct voltage and current? Yes. These types of LED lights are designed to work with 4.5V of non-rechargeable batteries and will provide light down to about 3V. By staying just under the engineered voltage for the LED lamps, you are not likely to overdrive them and still achieve an acceptable amount of light.