Consider a random matrix $X$ whose entries are i.i.d.\ in the cells of a Young diagram (its `shape') and zero elsewhere. When the shape is the dilation by a factor $N$ of a fixed Young diagram $\lambda$, the Wishart-type matrix $XX^*$ (suitably rescaled) has, as $N\to\infty$, a limiting spectral distribution $F^{\lambda}$ characterised by its moments. These moments enumerate $\lambda$-plane trees, a class of directed plane trees with vertex labelling compatible with $\lambda$, for which we provide explicit enumerative formulae. We show that one cannot `hear the shape of a random matrix', in the sense that there exist distinct Young diagrams yielding the same limiting spectral distribution. We establish that the classes of `isospectral' Young diagrams are those with the same diagonal profile.