Packing and covering random trees

Abstract

We study the sizes of optimal packings and coverings of conditioned branching processes by balls with growing radii. We find a universal limiting constant, and thereby establish universality of packing and covering numbers across different scales and different offspring distributions. To be precise, for $μ$ an offspring distribution with mean $1$ and finite variance $σ^2$ we let $T_n$ be a Bienaymé tree with offspring distribution $μ$ conditioned to have size $n$. We prove that the packing number (resp. covering number) of $T_n$ with balls of radii $1\ll γ_n \ll n^{1/2}$ is asymptotically equivalent to $\mathfrak{C} n /(σ^2 γ_n^2)$ (resp. $\mathfrak{C} n /(4σ^2 γ_n^2)$), for a universal constant $\mathfrak{C}$. We also show that the order of limits can be exchanged and that the same limit occurs when we first consider the limiting covering number of $T_n$ by balls of fixed radius $R$, and thereafter let $R$ tend to $\infty$. We then extend these results to the Brownian continuum random tree conditioned to have mass $1$. For this model, we prove that the packing (resp. covering) number with balls of radii $\varepsilon$ are asymptotically equivalent to $\mathfrak{C}/\varepsilon^2$ (resp. $\mathfrak{C}/(4\varepsilon^2)$), for the same constant $\mathfrak{C}$, in contrast with a conjecture by Aldous. Finally, our results answer a question by Devroye, Eide and Pralat on the asymptotic burning number of mean $1$ finite variance Bienaymé trees of size $n$, by showing that it lies between $(c+o(1)) n^{1/3}$ and $(2c+o(1))n^{1/3}$ with high probability, for a constant $c$ that only depends on the offspring variance.