We present PTIME language-preserving techniques for the reduction of non-deterministic tree automata, both for the case of finite trees and for infinite trees. Our techniques are based on new transition removing and state merging results, which rely on binary relations that compare the downward and upward behaviours of states in the automaton. We use downward/upward simulation preorders and the more general but EXPTIME-complete trace inclusion relations, for which we introduce good under-approximations computable in polynomial time. We provide a complete picture of combinations of downward and upward simulation/trace inclusions which can be used in our reduction techniques. We define an algorithm that puts together all the reduction results found for finite trees, and implemented it under the name minotaut, a tool built on top of the well-known tree automata library libvata. We tested minotaut on large collections of automata from program verification provenience, as well as on different classes of randomly generated automata. Our algorithm yields substantially smaller and sparser automata than all previously known reduction techniques, and it is still fast enough to handle large instances. Taking reduction of automata on finite trees one step further, we then introduce saturation, a technique that consists of adding new transitions to an automaton while preserving its language. We implemented this technique on minotaut and we show how it can make subsequent state-merge and transition-removal operations more effective. Thus we obtain a PTIME algorithm that reduces the number of states of tree automata even more than before. Additionally, we explore how minotaut alone can play an important role when performing hard operations like complementation, allowing to obtain smaller complement automata and at lower computation times overall. We then show how saturation can extend this contribution even further. An overview of the tool, highlighting some of its implementation features, is presented as well.