A conceptual link between trees and Polish spaces Could somebody explain to me why trees are so relevant for the study of Polish spaces and descriptive set theory?
I still do not get the proper connection (...and when I think I got it – see the Lusin scheme – I find out my intuition failed).
Any feedback or answer is most welcome. 
Thank you for your time.
 A: In the comment section people have given a very good, fundamental reason why trees are so important to descriptive set theory, namely, the universality of the sequence spaces $A^\omega$ for countable $A$. I would like to add something to these comments, and perhaps this might help you in getting some new intuition.
How do you describe a $G_\delta$ (i.e., $\Pi_2^0$) set in general? You just take a countable family $\{U_i\}_{i\in\omega}$ of open sets and take the intersection: $\bigcap_i U_i$. Now if you want to climb one more step in the Borel hierarchy and take a look to the general shape of a $G_{\delta\sigma}$ set, you must take a (doubly indexed) countable family of open sets $\{U_{ij}\}_{i,j\in\omega}$ and then
$$
\bigcup_j\bigcap_i U_{ij}.
$$
is a general  $G_{\delta\sigma}$ set. When getting into higher (finite) levels of the hierarchy you will end up with a family of open sets indexed by finite sequences of the natural numbers. I believe that one great insight was to realize that the set of indices had a topological structure of their own, and this is so fundamental that it finally turned out that this topology underlies the structure of all Borel (and analytic, because of the Suslin operation) sets in Polish spaces. As an example of this is the following theorem: for every Borel set $B$ in a Polish space there is a tree $T$ on $\omega$ and a continuous bijection from the (closed) set $[T]$ of infinite branches of $T$ onto $B$.
