Motivated by the study of the Ising model on tree-like random graphs with random boundary conditions, we define a random artificial boundary for a supercritical branching process. We show that the phase transition for the root magnetisation on the Ising model with this type of boundary is equivalent to that of the "+"-boundary conditions for a certain inhomogeneous tree model: the Pruned tree. We introduce the model and show criteria for the root magnetisation in terms of the 3-capacity of the Pruned tree.