# Log-normal distribution and neural connections

I have some data from neuroscience, to be more precise a set of data which represents the connections between brain cells (so called "axons"). These connections form a binary tree as depicted below. Interestingly when I look at the lengths of these edges (i.e. axons) at each depth, the lengths are lognormally distributed. I.e. the black edges length has a lognormal distribution, the orange edges length has a lognormal distribution and so on.

More then that, even if I look at the distribution of the lengths of all edges from all depths, still I do observe a lognormal distribution. I dont add the edges up but just again consider each edges length as the Random Variables Value and plot its distribution.

So my question is: How can it be explained analytically, that from lognormals at each depth also a lognormal over all depths arise? The Log-normal and Gamma distributions can easily be made to look similar. They are defined on the same support $(0, \infty)$, and can be "mound-shaped" given the appropriate parameters. Is it possible that a Gamma distribution can model your data?
If so, there is a straightforward answer to your question. The Gamma distribution has the additive property, that is, if $$X_i \sim \text{Gamma}(\alpha_i, \beta), \text{ for } i = 1, 2, ..., n,$$ then $$\sum_{i = 1}^n X_i \sim \text{Gamma}(\sum_{i = 1}^n \alpha_i, \beta),$$ which can be proven using characteristic functions. Perhaps this explains the interesting behavior you see.