We consider random matrices whose shape is the dilation $Nλ$ of a self-conjugate Young diagram $λ$. In the large-$N$ limit, the empirical distribution of the squared singular values converges almost surely to a probability distribution $F^λ$. The moments of $F^λ$ enumerate two combinatorial objects: $λ$-plane trees and $λ$-Dyck paths, which we introduce and show to be in bijection. We also prove that the distribution $F^λ$ is algebraic, in the sense of Rao and Edelman. In the case of fat hook shapes we provide explicit formulae for $F^λ$ and we express it as a free convolution of two measures involving a Marchenko-Pastur and a Bernoulli distribution.