For a given positive integer , does there exist a weighted tree with graph 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.
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.