Multi-task learning commonly encounters competition for resources among tasks, specifically when model capacity is limited. This challenge motivates models which allow control over the relative importance of tasks and total compute cost during inference time. In this work, we propose such a controllable multi-task network that dynamically adjusts its architecture and weights to match the desired task preference as well as the resource constraints. In contrast to the existing dynamic multi-task approaches that adjust only the weights within a fixed architecture, our approach affords the flexibility to dynamically control the total computational cost and match the user-preferred task importance better. We propose a disentangled training of two hyper networks, by exploiting task affinity and a novel branching regularized loss, to take input preferences and accordingly predict tree-structured models with adapted weights. Experiments on three multi-task benchmarks, namely PASCAL-Context, NYU-v2, and CIFAR-100, show the efficacy of our approach. Project page is available at https://www.nec-labs.com/-mas/DYMU.