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Random trees conditioned on the number of vertices and leaves

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If you have a question about this talk, please contact Perla Sousi.

I will talk about Galton-Watson trees conditioned on both the total number of vertices $n$ and the number of leaves $k$. Both $k$ and $n$ are assumed to grow to infinity and $k = \alpha n + O(1)$, with $\alpha \in (0, 1)$. Assuming the exponential decay of the offspring distribution, I show that the rescaled random tree converges in distribution to Aldous’ Continuum Random Tree with respect to the Gromov-Hausdorff topology. The rescaling depends on a parameter $\sigma^2$ which can be calculated explicitly. Additionally, I will describe the limit of the degree sequence for the conditioned trees.

This talk is part of the Probability series.

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