My question is very imprecise, as I know very little about descriptive set theory.

In a problem I am thinking about I have a family of well-founded trees (finite sequences on $\cup_n X^n$ closed under taking predecessors, such that every path has a maximal element) on a Polish space (separable, completely metrizable) space. These trees have several additional proprieties/constraints. I am trying to construct for each $\alpha<\omega_1$ a tree that has order at least $\alpha$. The order of the tree is defined inductively by removing maximal elements until nothing is left. The least ordinal for which this happens is the order of the tree.

My questions, as I said very imprecise, is what properties of a tree imply that the order is larger than a prescribed ordinal? Are there any standard techniques of constructing such trees? I know this is a long shot as I have several other constraints on these trees, but if there is some relevant literature where I can start looking, that would be appreciated. I know of the standard reference of Kechris book, but I couldn't find something that I can use (or perhaps I did not know where to look in that book).

Thank you and if you think this question is too vague, please just delete it.