For a tree with a given root, let $d_i$ be the number of vertices at distance $i$ from the root. Given the distance distribution $d =[d_1, d_2, \cdots, d_k]$ ($k$ is the diameter), how many (non-isomorphic) trees can we construct with the same root?
Further, if we restrict the maximum degree of each vertex to be some constant $c$, how does the count change? Even more, how many connected simple graphs can we construct such that at least one vertex has the distance distribution $d$.
This is a problem in mathematical chemistry or precisely Inverse QSAR. I would chose $c$ to be 4 (maximum valency of atoms). Some function of distance distribution $d$ are used as "descriptor" of molecules (topological indices) and correlation are established between these descriptors and bioactivity. Now the problem is to design molecules (graph representations) that would correspond to the descriptors.