Abstract: We consider regular critical multitype Bienaym\'e trees $\mathcal{T}_n$, conditioned on the value $n$ of a linear combination of the number of vertices of a given type. We prove that the scaling limit of the sequence $(\mathcal{T}_n)_{n \geq 1}$ is the Brownian Continuum Random Tree, and establish strong non-asymptotic tail bounds for the height of $\mathcal{T}_n$. Our main tool is a flattening operation on multitype trees, as well as sharp estimates on the structure of monotype trees with a given sequence of degrees.
Joint work with Louigi Addario-Berry, Benedikt Stufler and Paul Thévenin.