I encountered this exercice on trees :
Suppose $T$ is an Aronszajn tree on $\omega_1$. Find a partial ordering $\mathbb{P}$ and a family $\mathcal{F}=\{D_\alpha:\alpha<\omega_1\}$ dense subsets of $\mathbb{P}$ such that if $G$ is $\mathcal{F}$-generic, then $\bigcup G : T\to\omega$ is such that the preimage of each $n$ is an antichain of $T$, i.e., $G$ yields a partition of $T$ into countably-many antichains.
I would appreciate someone opinion on my line of thoughts:
First, I need to find a poset $\mathbb{P}$, and a family $\cal F$ such that if a there is $G$ $\cal F$-generic, then $T$ is special. So $\mathbb{P}$ must be closely related to $T$. why not take $T$ itself then?
Let $\mathcal{A}$ be the family of all maximal antichain of $\mathbb{P}$, and for any $A\in\mathcal{A}$, let $$ D_A =\{ p, \ p\leq q \text{ for some }q\in A\}$$ We know that for any maximal antichain $A$, $D_A$ is dense open. Therefore we define $$ \mathcal{F} = \{ D_A, \ A\in\mathcal{A}\}$$
Now suppose we have $G$, $\mathcal{F}$-generic. Then $G$ intersect each $D_A$. I feel that I'm getting close, but I can't conclude. Any help?
Your approach is the wrong way to think about it, since the generic subset won't code a function from $T$ to $\omega$. Because we're trying to approximate a function from $T$ to $\omega$ (with some special properties), the natural poset to use is just finite approximations to such a function.
In particular, we can take $\mathbb{P}$ to be the set of finite partial functions from $T$ to $\omega$ such that the pre-images of each $n\in\omega$ are antichains. And as usual for functions, we can order them by reverse inclusion: $p\leqslant q\in\mathbb{P}$ iff $p\supseteq q$.
To ensure the generic $G$ codes a function $\bigcup G$ defined on all of $T$, have $D_t$ basically state "$t$ in in my domain". Explicitly, you can consider $D_t= \{ p\in\mathbb{P}:t\in\mathrm{dom}(p) \} \text{.}$ It's not difficult to show that each $D_t$ is dense in $T$, and the preimages of $\bigcup G$ of each $n\in\omega$ still remain antichains.