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.

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1999-05-26