Short leaf-to-leaf path lengths in 1–3 trees

Establish whether there exist a constant \(\alpha>0\) and a function \(N=N(n)\) tending to infinity with \(n\) such that every 1–3 tree of order \(n\) contains at least \(\alpha N\) distinct leaf-to-leaf path lengths between 0 and \(N\).

Background

The paper studies the number of distinct lengths of paths whose endpoints are leaves of a tree. A 1–3 tree is a tree in which every vertex has degree either 1 or 3. The conjecture asks for a positive-density set of attainable leaf-to-leaf path lengths in an initial interval [0,N][0,N], where the interval length itself grows with the order of the tree.

The authors prove a weaker result: under the more general hypothesis that a tree has no vertices of degree 2 and has sufficiently large diameter, it contains on the order of N2/3N^{2/3} distinct leaf-to-leaf path lengths in a bounded interval. Thus, the linear-in-NN conclusion of the conjecture remains unresolved.

References

They conjectured that for 1--3 trees, one can find path lengths which are dense in an interval of the form $[0,N]$. There exist a constant $\alpha >0$ and a function $N = N(n)$ tending to infinity as n \rightarrow \infty such that every 1--3 tree of order $n$ contains at least $\alpha N$ distinct leaf-to-leaf path lengths between 0 and $N$.

— Leaf-to-leaf paths of many lengths  (2501.18540 - Braccio et al., 30 Jan 2025) in Conjecture 6.4 of Narins, Pokrovskiy and Szabó, stated in Section 1; labelled Conjecture \ref{conj:smalllengths}