I'm a software developer by trade and one of the things I did today was recursively generate some dummy data for a tree. I used two parameters in my recursive algorithm: number of levels and number of items per level. From doing things like this in the past I know it's a quick and easy way to produce a lot of dummy data but today I realized that I couldn't predict how much would be produced. I'm sure there must be an equation that exactly links the input variables with the number of items in the tree.
I've tried various equations involving powers and factorials etc but none give the correct result. Plotting the data for three levels in Excel and fitting a curve gave a equation of approximately y=3x^2.5 but it's a fairly poor fit.
So my question is: how would a mathematician go about finding the equation for this data?
Please note, I tagged this homework because I didn't know what else to tag it and that seemed the best fit. Perhaps there should be a "personal challenge" tag :-)
Sample Tree - 2 Levels, 2 Items
Root - 0 - - 0.0 - - 0.1 - 1 - - 1.0 - - 1.1
Total: 7 Items
Sample Data
Levels Items Count 1 10 11 2 1 3 2 2 7 2 3 13 2 4 21 3 1 4 3 2 15 3 3 40 3 4 85 3 5 156 3 6 259 3 7 400 3 8 585 3 9 820 3 10 1111 5 1 6 5 2 63 5 3 364 5 4 1365 5 10 111111