## Taylor's Condition

For a given Positive Integer , does there exist a Weighted Tree with Vertices whose paths have weights 1, 2, ..., , where is a Binomial Coefficient? Taylor showed that no such Tree can exist unless it is a Perfect Square or a Perfect Square plus 2. No such Trees are known except , 3, 4, and 6.

References

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1985.

Leech, J. Another Tree Labeling Problem.'' Amer. Math. Monthly 82, 923-925, 1975.

Taylor, H. Odd Path Sums in an Edge-Labeled Tree.'' Math. Mag. 50, 258-259, 1977.