Random trees have height $O(\sqrt{n})$

Abstract

We obtain new non-asymptotic tail bounds for the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaymé trees (the family trees of branching processes), in the process settling three conjectures of Janson (2012) and answering several other questions from the literature. Moreover, we define a partial ordering on degree sequences and show that it induces a stochastic ordering on the heights of uniformly random trees with given degree sequences. The latter result can also be used to show that sub-binary random trees are stochastically the tallest trees with a given number of vertices and leaves (and thus that random binary trees are the stochastically tallest random homeomorphically irreducible trees with a given number of vertices). Our proofs are based in part on the bijection between trees and sequences introduced by Foata and Fuchs (1970), which can be recast to provide a line-breaking construction of random trees with given vertex degrees as shown in Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin (2023).

Figures (6)

• Figure 1: In the tree $\mathrm{t}$, there are 5 vertices with no children, two vertices with two children and one vertex with 3 children, so for $\mu=(\mu_k,k\geq 0)$ a probability distribution on ${\mathbb N}$, we have that $\mathbf P(\mathrm{T}_\mu=\mathrm{t})=\mu_0^5\mu_2^2\mu_3$.
• Figure 2: This figure illustrates the bijection and the sequential construction. In this example, we have $\pi(2)=5$ since $\mathrm{t}_5$ is the first tree in the sequence containing vertex $2$. We also have $\rho(2)=4$, since $3$ is the minimal $k$ such that $\sum_{1\leq j\leq k}(d_{i(j)}-1)$ is at least $2$ and $\mathrm{t}_4$ is the first tree in the sequence to contain vertices $\{i(1),i(2),i(3)\}=\{4,3,1\}$.
• Figure 3: Left: a tree $\mathrm{t}$. Right: the tree $\mathrm{t}^-$ obtained from $\mathrm{t}$ by suppressing degree-one vertices. Considering the vertices in the order $(1,2,5,6,9)$, the corresponding labeled composition of the set $\{3,4,7,8\}$ of degree-one vertices is $(),(),(8),(7,3),(4)$.
• Figure 4: We show a subset of the equivalence class of tree $\mathrm{t}$. In total, $\mathrm{t}$ is equivalent to $2^4$ trees: to specify a tree $\mathrm{t}'$ such that $\mathrm{t}'\sim \mathrm{t}$ we must choose, for each element in $\{6,8,9\}$, whether its parent in $\mathrm{t}'$ is $1$ or $2$, and we must choose whether or not to swap the labels of $1$ and $2$.
• Figure 5: We show a subset of the equivalence class of tree $\mathrm{t}$. In total, $\mathrm{t}$ is equivalent to $2^3$ trees: to specify a tree $\mathrm{t}'$ such that $\mathrm{t}'\sim \mathrm{t}$ we must choose, for each element in $\{4,5\}$, whether its parent in $\mathrm{t}'$ is $1$ or $2$, and we must choose whether or not to swap the labels of $1$ and $2$.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem : blanc, Theorem 8(a)
• Proposition 8
• proof : Proof of Proposition \ref{['prop:planetrees']}