More than 25 years ago John Leech posed the following beautiful problem: find, whenever possible, trees on n vertices with positive weights on the edges, such that the (n/2) weighted distances among the n vertices are exactly the number 1, 2, 3, ..., (n/2).

Herbert Taylor gave a proof restricting the number of vertices on which Leech trees can exist to n2 and n2+2. We prove two theorems, giving restrictions on the longest path and maximum degree in Leech trees. We also prove a computer search showing that there is no Leech tree for n =9, 11.