Scaling limit of graph classes through split decomposition

We prove that Aldous’ Brownian CRT is the scaling limit, with respect to the Gromov–Prokhorov topology, of uniform random graphs in each of the three following families of graphs: distance-hereditary graphs, $2$-connected distance-hereditary graphs and $3$-leaf power graphs. Our approach is based on the split decomposition and on analytic combinatorics.