We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire space ω ω 1 1. First, we show that none of these classes have the Baire property (unless they are empty). Moreover, under V = L, (a) the class of Aronszajn and Suslin trees is Π 1 1-complete, (b) the class of special Aronszajn trees is Σ 1 1-complete, and (c) the class of Kurepa trees is Π 1 2-complete. We achieve these results by finding nicely definable reductions that map subsets X of ω1 to trees TX so that TX is in a given tree class T if and only if X is stationary/non-stationary (depending on the class T). Finally, we present models of CH where these classes have lower projective complexity. 1 and 2 ω 1. Basic open sets correspond to countable partial functions, which, in turn, give rise to an ω 1-Borel structure on ω ω 1 1 , 2 ω 1 and so P(ω 1) as well. This allows us to measure the complexity of subsets of ω ω 1 1 or, equivalently, of families of natural combinatorial structures on ω 1. In this paper, we will focus on models of CH, i.e., 2 ℵ 0 = ℵ 1. This is a fairly natural assumption in