  Suppose $G$ is a tree. Graham's "Tree Reconstruction Conjecture" states that
$G$ is uniquely determined by the integer sequence $|G|$, $|L(G)|$,
$|L(L(G))|$, $|L(L(L(G)))|$, $\ldots$, where $L(H)$ denotes the line graph of
the graph $H$. Little is known about this question apart from a few simple
observations. We show that the number of trees on $n$ vertices which can be
distinguished by their associated integer sequences is $e^{\Omega((\log
n)^{3/2})}$. The proof strategy involves constructing a large collection of
caterpillar graphs using partitions arising from the Prouhet-Tarry-Escott
problem.
