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.
