Critical trees are neither too short nor too fat

TL;DR

This paper characterizes how the height and width of critical Bienaymé trees conditioned on size behave. Using lattice-path encodings, the Foata–Fuchs bijection, and a careful analysis in both the generic critical and Cauchy-domain regimes, it proves that Height(T_n) grows faster than any constant multiple of ln n while Width(T_n) grows sub-linearly in n, with bounds proven to be optimal in general. In the Cauchy-attraction regime, the authors obtain precise asymptotics for key scaling sequences a_n, b_n, and h_n, and show Height(T_n) ∼ h_n and Width(T_n) ∼ b_n in probability, yielding a universal relation Height(T_n) Width(T_n) = O_P(n ln n). The work also demonstrates that, under certain constructions, trees can be simultaneously very short and very wide, highlighting the non-universality of some tails and posing open questions about further asymptotic regimes. Overall, the paper advances understanding of how size-conditioning interacts with criticality to shape the extreme geometry of Bienaymé trees, with implications for related random-map models and probabilistic combinatorics.

Abstract

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

Figures (4)

• Figure 1: Left: A walk $\mathbf{s}=(s_0,\ldots,s_{10})$ with $s_{10}=-1$. Centre: the Vervaat transform $\mathcal{V}(\mathbf{s})$. Right: the plane tree $\mathrm{t}$ with $\mathsf{W}^{\mathsf{bfs}}(\mathrm{t})=\mathcal{V}(\mathbf{s})$.
• Figure 2: Left: a simulation of $(S_{i})_{0 \leq i \leq n-1}$ for $n=100000$, where $(S_i,i \ge 0)$ is the random walk defined in Section \ref{['sec:scalingconstants']} with $\mu_n\sim \frac{c}{n^{2} \ln(n)^{2}}$. Right: the associated discrete excursion $Z^{(n)}$ obtained by performing the Vervaat transform on $(S_0,S_1,\ldots,S_{n-1},-1)$.
• Figure 3: The random walk $(S_i : 0\leq i < n)$ and the associated tree $\mathrm{T}'_{ n}$. In red, the $\vert S_{n-1} \vert$ first excursions of $S$ that encode the trees grafted above the vertex with maximal degree $v'_n$. In blue, the forest $\mathcal{F}^{n}_{\mathrm{R}}$ made of the trees grafted on the children of $\llbracket \varnothing, v'_{n}\llbracket$ on the right of $\llbracket \varnothing, v'_{n}\llbracket$. In this example, $\mathsf{H}(\mathcal{F}^{n}_{\mathrm{L}})=2$ and $\mathsf{H}(\mathcal{F}^{n}_{\mathrm{R}})=4$.
• Figure 4: A sequence $\mathrm{v}=(v_1,\ldots,v_8)$ and the tree $\mathrm{t}(\mathrm{v})$ that it encodes. For $i=1,\dots, 5$, the colour of the edges on the path $P_i$ in $\mathrm{t}(\mathrm{v})$ equals the colour underlining entries $v_{j(i-1)},\ldots,v_{j(i)-1}$ of $\mathrm{v}$. Figure adapted from cayley_us.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Proposition 2.1
• Remark 2.2
• Theorem 3.1: Theorem 21 in KR19
• Proposition 3.2: Proposition 24 in KR19
• Corollary 3.3
• Lemma 3.4