On Leech’s Tree Labelling Problem

Abstract

J. Leech has posed the following problem: For each integer $n$, what is the greatest integer $N$ such that there exists a labelled tree with $n$ nodes in which the distance between the pairs of nodes include the consecutive values $1,2,\dots ,N$? With the help of a computer, we get $B(n)$ (the number $N$ for branched trees) for $2\le n\le 10$ and lower bounds of $B(11)$ and $B(12)$. We also get $U(n)$ (the number $N$ for unbranched trees) for $2\le n\le 11$ independently, confirming some results gotten by J. Leech.