about the previous topic for perfect distance forests, two references that might be helpful for anyone interested,

References:

[1] B.Calhoun, J. Polhill, Perfect distance forests, Australian Journal of Combinatorics, vol. 42(2008), p. 211-222.(NOT AVAILABLE ONLINE)

[2] W.Calhoun, K. Ferland, L. Lister and J. Polhill, Minimal distinct distance trees, J.Combin. Math. Combin. Computing vol. 61(2007) p. 33-57.(AVAILABLE HERE)

so leech trees of order 6 or smaller are known. For order 9,11 or 16 there no Leech trees either leaving 18 the smallest open case. Szekely has conjectured that no additional Leech trees exist but this is still open. Another generalization can be studied in a different domain, .

Let be a tree of vertices and let . We say that is a modular Leech tree if there exists an edge weighting function such that each of the paths within has a distinct weight from with the sums taken modulo . We call such an edge weighting function a  -Leech labeling. Since the pathweights are all distinct, the function induces a bijection between the paths of and the elements of the group . We use to refer to this bijection as well.

Note that a perfect distance tree as defined previously of order is also a modular Leech tree over in which none of the path sums, before applying the mod operation, have a weight greater than . Thus Leech’s original five examples provide us with five modular Leech trees. For more read here.