Székely et al. showed computationally that there are no such trees with
and 11. They also showed that if there is such a tree on vertices then the maximum
vertex degree is at most and that there is no path of length larger
than .
They conjecture that there are only finitely many such trees.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1985.Leech,
J. "Another Tree Labeling Problem." Amer. Math. Monthly82,
923-925, 1975.Székely, L. A. "Programs for Leech Tree
of Nine Nodes and Eleven Nodes." http://www.math.sc.edu/~szekely/leechtree/.Székely,
L. A.; Wang, H.; and Zhang, Y. "Some Non-Existence Results on Leech Trees."
Bull. Inst. Combin. Appl.44, 37-45, 2005.Taylor, H. "Odd
Path Sums in an Edge-Labeled Tree." Math. Mag.50, 258-259, 1977.