UCLA

A Leech tree of order n is a tree on n vertices with positive integer edge weights whose n(n-1)/2 pairwise path-weights are exactly 1,2,...,n(n-1)/2. Leech (1975) found five, of orders 2,3,4,4,6; Taylor (1977) showed that the order must be a square or a square plus two; earlier searches excluded the admissible orders 9, 11 and 16, leaving 18 as the smallest open order. We prove that there is no Leech tree on 18 vertices, by an exhaustive generation of forced forests over all 123,867 trees of that order. The search visits 59,779,854,336 nodes and returns no survivor; its per-level counts were reproduced by independent runs and by a clean-room implementation written from the algorithm description alone. The same conclusion was reached independently and concurrently by Ghodsi, through a different reduction and a different trusted base. We also prove that no spider of order n>=5 is a Leech tree, by a finite exact analysis of the largest distances together with independently implemented checks. For the next admissible order n=25 we prove a structural bound rather than a non-existence result: deleting a leaf forces a block of consecutive rooted depths whose pairs are too crowded for the block to be long, so that diam(T-l)>=281 for every leaf, unconditionally. Four finite certified searches raise this to 285, reducing order 25 to fourteen values of one explicit finite normal form; we make no claim that order 25 is settled. Variants of the same engines settle neighbouring questions: M(11)=60 and M(12)=77 for the minimal distinct-distance trees of Calhoun et al., each with exactly two minimal trees; there is no modular Leech tree of order 9 or 11, while order 5 does admit them, contrary to a statement in the literature; and there are exactly six leaf-Leech trees with six leaves.