Tight universal bounds on the height times the width of random trees

TL;DR

This work delivers a universal, non-asymptotic bound on the product of height and width for a broad class of random trees, showing ${\mathsf{Wd}}(T){\mathsf{Ht}}(T)=O(n\log n)$ in probability with explicit tail decay. The authors develop a spine-based framework, encode trees via depth-first and breadth-first walks, and analyze exchangeable bridges and their Vervaat transforms to relate width to spinal weights. Central to the approach are five propositions tied to line-like structure, width-spine relationships, and symmetry via shuffling, which together yield a unified bound covering Bienaymé, simply generated, and fixed-degree-sequence trees. The results settle an open question, establish tight growth order, and open avenues for refined tail estimates and classification of offspring distributions by height-width scaling regimes, with potential extensions to exponential tail improvements. The methods combine combinatorial encodings with second-moment techniques and symmetry arguments to achieve assumption-free, uniform control over a broad family of random trees.

Abstract

We obtain assumption-free, non-asymptotic, uniform bounds on the product of the height and the width of uniformly random trees with a given degree sequence, conditioned Bienaymé trees and simply generated trees. We show that for a tree of size $n$, this product is $O(n \log n)$ in probability, answering a question by Addario-Berry (2019). The order of this bound is tight in this generality.

Figures (9)

• Figure 1: A depiction of a plane tree $t$. The type of $t$ is $\mathtt{n}(t)=(5,2,2,1,0,0,\dots)$. The height of $t$ is ${\mathsf{Ht}}(t)=4$ (realised by vertices $1121$ and $1122$) and its width is $\mathsf{Wd}(t)=4$ (realised by the generation at distance $3$ from the root). The right spinal weight of vertex $112$ in $t$ is $\mathrm{S}_{112}^{\mathrm{d}}(t)=\#\{113,12\}=2$ and the left spinal weight of vertex $112$ is $\mathrm{S}_{112}^{\mathrm{g}}(t)=\#\{111\}=1$. Its spinal weight is $\mathrm{S}_{112}(t)=\mathrm{S}_{112}^{\mathrm{d}}(t)+\mathrm{S}_{112}^{\mathrm{g}}(t)=3$. The second order height of $t$ is ${\mathsf{Ht}}^{(2)}(t)=2$, where the maximum of $\min(|u|-|u\wedge v|,|v|-|u\wedge v|)$ is realised when either $\{u,v\}=\{1121,121\}$ or $\{u,v\}=\{1122,121\}$.
• Figure 2: A depiction of the cyclic shift of $t$ along the ancestral line of $v$ for $i=3$. The vertices $v, v_3,\varnothing$ in $t$ are marked, and so are their images under $\psi$ in $\psi(t)$. The same holds for four arbitrary vertices $w,x,y,z$ in $t$.
• Figure 3: A tree $t$ together with its depth-first walk $X^\mathrm{df}(t)$ and breadth-first walk $X^\mathrm{bf}(t)$. The depth-first exploration of $t$ is given by $u^\mathrm{df}(t)=(\varnothing,1, 11, 111, 112,1121,1122,113,12,121)$ and its breadth-first exploration is given by $u^\mathrm{df}(t)=(\varnothing,1, 11,12,111,112,113,121,1121,1122)$.
• Figure 4: A bridge $\mathtt{y}$ and its Vervaat transform $V(\mathtt{y})$.
• Figure 5: A depiction of a tree $t$ and the tree $t^{(9)}$ that is obtained by taking all subtrees stemming from $u^{\mathrm{df}}_9(t)$ and from younger siblings of $u^{\mathrm{df}}_9(t)$ itself and its ancestors and attaching these trees to a root vertex.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5