Graham's tree reconstruction conjecture

Let $G=(V,E)$ be a simple undirected graph. A tree is an acyclic, connected graph. Its line graph $L(G)$ has vertex set $E$, with distinct $e,f\nin E$ adjacent exactly when $e$ and $f$ are incident in $G$. Define the iterated line graphs by

$$L^{(0)}(G)=G,$$

and

$$L^{(j+1)}(G)=L(L^{(j)}(G))\qquad (j\geq 1).$$

Graham's tree reconstruction conjecture. For each sequence of natural numbers $a_0,a_1,a_2,\ldots$, all the conditions $|L^{(j)}(G)|=a_j$ for $j\geq 0$ are satisfied by at most one tree $G$.

The conjecture asks whether the sequence of sizes of a tree and all its iterated line graphs uniquely determines the tree. Apart from a few simple observations, little is known; the paper gives lower bounds on the number of trees distinguishable by these sequences but does not resolve the conjecture.