In this work, we introduce a new decision tree ensemble representation framework: instead of using a graph model we transform each tree into a well-known polynomial form. We apply the new representation to three tasks: theoretical analysis, model reduction, and interpretation. The polynomial form of a tree ensemble allows a straightforward interpretation of the original model. In our experiments, it shows comparable results with state-of-the-art interpretation techniques. Another application of the framework is the ensemble-wise pruning: we can drop monomials from the polynomial, based on train data statistics. This way we reduce the model size up to 3 times without loss of its quality. It is possible to show the equivalence of tree shape classes that share the same polynomial. This fact gives us the ability to train a model in one tree's shape and exploit it in another, which is easier for computation or interpretation. We formulate a problem statement for optimal tree ensemble translation from one form to another and build a greedy solution to this problem.