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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

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Ration distribution is a non-trivial problem for two random variables with unknown distributions. Though there are a lot of solutions for simple cases, for more see here Ratio distribution. Also for small random tree networks, see this paper "Variability of collective dynamics in random tree networks of strongly coupled stochastic excitable elements"

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