# What is the ratio of the number of the leaves to the total number of nodes in a random tree?

I generate the random trees using a Galton-Watson process in the following way:

• The maximum number of Generation is G=4
• The minimum and the maximum number of branchings are zero and five (0-5) respectively.
• Having zero branching is allowed only after $G \geq 2$. Otherwise, there is a high probability that I will end up with trees with only one node.
• The nodes that have no branching either in the last generation (G=4) or in the inner generations (G<4) are called heminodes and the sum of all these nodes is called $\mathcal{H}$
• The total number of nodes in each random tree is called $\mathcal{N}$

I do the above simulation for $10^6$ times, The following is the probability distribution of $R= \frac{\mathcal{H}}{\mathcal{N}}$.

The probability distribution of $R= \frac{\mathcal{H}}{\mathcal{N}}$ (click to see the figure please)

Is there any mathematical theory that could give the distribution for any given number of generations and branching?

For the importance of this ratio, I recommend having a look at our recent paper. "Emergent stochastic oscillations and signal detection in tree networks of excitable elements" https://www.nature.com/articles/s41598-017-04193-8