On Leech’s Tree Labelling Problem

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 \(1,2,\ldots,N\)? With the help of a computer, we get \(B(n)\) (the number \(N\) for branched trees) for \(2 \leq n \leq 10\) and lower bounds of \(B(11)\) and \(B(12)\). We also get \(U(n)\) (the number \(N\) for unbranched trees) for \(2 \leq n \leq 11\) independently, confirming some results gotten by J. Leech.