Abstract: We consider advection-diffusion equations posed on a tree with suitable transmission conditions at the junctions. We prove that the system is null-controllable using a control which is localized on the exterior nodes. Moreover, we study the asymptotic behavior of the cost of the null-controllability as the diffusivity parameter vanishes: we show that it decays for a sufficiently large time and explodes for short times. These results have been obtained in collaboration with J. A. Bárcena-Petisco, M. Cavalcante, G. M. Coclite, and E. Zuazua.