Leech trees beyond six vertices

A Leech tree is a tree with positive integer edge lengths whose distances between pairs of vertices are exactly $1, 2, 3, \dots$ , each appearing once. John Leech found five such trees in 1975, the largest with six vertices, and none has been found since. The task is to find a new one, or to prove that there are no others.

Problem

Let $T$ be a tree on $n$ vertices whose edges carry positive integer weights, and let the distance between two vertices be the total weight of the path joining them. $T$ is a Leech tree if its $\binom{n}{2}$ pairwise distances are exactly $1, 2, \dots, \binom{n}{2}$ .

Task. Find a Leech tree with more than six vertices, or prove that none exists. By the results below, such a tree must have at least 18 vertices, and at least 25 if the 2026 computation for 18 vertices is correct.

Background

Leech introduced these trees in 1975 [1]; Epoch notes his motivation from circuit design, with edges as resistors of integer resistance. Taylor proved that the number of vertices of a Leech tree must be a perfect square or two more than a perfect square [2], which leaves $n = 9, 11, 16, 18, 25, 27, \dots$ as candidates beyond six. Since then the small candidates have been eliminated one by one with structural arguments and exhaustive computer searches [3].

What is known

Five Leech trees are known, on 2, 3, 4, 4 and 6 vertices, all found by Leech [1, 3].
The number of vertices must be $m^2$ or $m^2 + 2$ (Taylor, 1977) [2].
Orders 9 and 11 were excluded by Székely, Wang and Zhang (2005), and order 16 by Calhoun, Ferland, Lister and Polhill (2007), as summarised in [3].
In September 2026 Ghodsi posted a computer-assisted proof that no Leech tree has 18 vertices [3]. The reduction of every candidate to eight local configurations is checked in Lean 4; the exhaustive search that closes them, and its certificate checker, are not formalized, and the preprint has not been refereed. If it stands, the smallest open orders are 25 and 27.

What would count as progress

An explicit weighted tree, given as a list of edges with weights, whose pairwise distances are exactly $1, \dots, \binom{n}{2}$ ; checking takes $O(n^2)$ time.
Epoch scores submissions with its own verifier: a JSON list of weighted edges forming a tree on at least 18 vertices.
Excluding further orders, or proving that no Leech tree has more than six vertices, would settle the problem the other way; an independent check of the order-18 computation is also valuable.

References

J. Leech, Another tree labelling problem, Amer. Math. Monthly 82 (1975), 923–925. doi:10.1080/00029890.1975.11993981
H. Taylor, Odd path sums in an edge-labeled tree, Math. Magazine 50 (1977), 258–259. doi:10.1080/0025570X.1977.11976658
M. Ghodsi, Nonexistence of a Leech tree of order 18: a computer-assisted proof, 2026. arXiv:2609.20492

Source

Imported from FrontierMath: Open Problems, leech-trees, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.