We initiate the study of tree structures in the context of scenario-based robust optimization. Specifically, we study Binary Search Trees (BSTs) and Huffman coding, two fundamental techniques for efficiently managing and encoding data based on a known set of frequencies of keys. Given a number of distinct scenarios, each defined by a frequency distribution over the keys, our objective is to compute a single tree of best-possible performance, relative to any scenario. We consider, as performance metrics, the competitive ratio, which compares multiplicatively the cost of the solution to the tree of least cost among all scenarios, as well as the regret, which induces a similar, but additive comparison. For BSTs, we show that the problem is NP-hard across both metrics. We also obtain an optimal competitive ratio that is logarithmic in the number of scenarios. For Huffman Trees, we likewise prove NP-hardness, and we present an algorithm with logarithmic regret, which we prove to be near-optimal by showing a corresponding lower bound. Last, we give a polynomial-time algorithm for computing Pareto-optimal BSTs with respect to their regret, assuming scenarios defined by uniform distributions over the keys. This setting captures, in particular, the first study of fairness in the context of data structures. We provide an experimental evaluation of all algorithms. To this end, we also provide mixed integer linear program formulation for computing optimal trees.