Can there be an almost-special not-fully-special Aronszajn tree? Question. Can there be an Aronszajn tree $T$, such that no
c.c.c. forcing extension adds a cofinal branch to $T$, but there is an
$\omega_1$-preserving forcing extension adding a cofinal branch to
$T$?
To give some background, an Aronszajn tree is an
$\omega_1$-tree with no $\omega_1$-branches, and it is special,
if it is the union of countably many antichains. Any forcing
extension in which a special Aronszajn tree $T$ comes to have a cofinal branch must collapse
$\omega_1$, since the branch has size $\omega_1^V$ and will contain
at most one node from each of the antichains witnessing that $T$ is
special. In other words, a special Aronszajn tree must remain Aronszajn in any
$\omega_1$-preserving forcing extension, and this includes all
c.c.c. extensions. Meanwhile, any Aronszajn tree can be forced to
be special, and the specializing forcing is c.c.c. and absolute
(consisting of finite partial specializing functions). Thus, this
forcing remains c.c.c. in any extension in which $T$ remains
Aronszajn.
Concerning the question,


*

*A special Aronszajn tree satisfies the first requirement of the question, since we
cannot add a branch to it by c.c.c. forcing, but it doesn't
satisfy the second requirement, since no $\omega_1$-preserving extension
can add a cofinal branch. So special Aronszajn trees won't work.

*A Souslin tree satisfies the second requirement, but not the first, precisely because it is
itself c.c.c. and adds a branch through itself. So Souslin trees
won't work.
What is needed is a tree that is almost special, in the sense that
it remains Aronszajn in every c.c.c. extension, but not fully
special, in the sense that we can add a cofinal branch by some
(non-c.c.c.) $\omega_1$-preserving forcing.
I am interested in this question in part because the existence of
such a tree $T$ would provide an answer Arthur Fischer's question
on two
versions of absolutely ccc, because the forcing $\mathbb{P}$ to specialize $T$ is c.c.c. and would remain
c.c.c. in any c.c.c. extensions, since the tree would still be Aronszajn
there, but it would not be c.c.c. in the $\omega_1$-preserving
extension in which $T$ gains a cofinal branch.
 A: To supplement this with another example, it is also possible to construct a tree that is $\omega$-distributive and $S$-st-special (in Shelah's terminology) from $\Diamond^*(S^c)$ for some $S$ bi-stationary (in fact I think $\Diamond(S^c)$ works). It appears in Shelah's proper forcing book Chapter IX Lemma 3.9. This tree satisfies your requirement since forcing with the tree itself adds no countable sequences of ordinals, and no forcing preserving stationary of $S$ (and preserving $\omega_1$) can add a branch by $S$-st-specialness. 
$T$ is $S$-st-special if there exists $f: T\restriction S\to \omega_1$ such that $f(x)\in ht(x)$ and $x<_T y \rightarrow f(x)\neq f(y)$.
A: The answer is yes. 
Force a generic $\square(\omega_1)$ sequence. This poset, $S$, is $\sigma$-closed, so it doesn't collapse $\omega_1$.
By Todorcevic, in the $\omega_1$-tree obtained from the minimal walks through this sequence  every branch of length $\omega_1$ codes a thread through the $\square(\omega_1)$ sequence, and vise verse (and this is absolute upward as long as $\omega_1$ is uncountable). 
In particular, in the model obtained from forcing the generic $\square(\omega_1)$ sequence, this tree is Aronszajn and as long as we don't collapse $\omega_1$, we know that it's enough to check what kind of forcing can add a thread to a $\square(\omega_1)$ sequence in order to know which forcing posets can add a branch.  
It is well known that the threading forcing $T$ of this generic $\square(\omega_1)$ sequence is $\omega_1$-distributive, and actually $S\ast T$ has a dense $\sigma$-closed subset.
On the other hand, c.c.c. forcing can't thread a $\square(\omega_1)$ sequence: 
Let $\langle C_\alpha\mid\alpha < \omega_1\rangle$ be a $\square(\omega_1)$ sequence, and assume that there is some c.c.c. forcing threading it. Let $D$ be a name for this thread (so $D$ is forced to be a club in $\omega_1$). 
By the c.c.c. there is some $D^\prime \subset D$ a club in the "ground model" (meaning before the c.c.c. forcing). 
Now, if $\alpha < \beta$ in $\text{acc } D^\prime$, then $D^\prime \cap \beta \subset C_\beta$ and therefore $\alpha \in \text{acc }C_\beta$, and by the coherence $C_\alpha = C_\beta \cap \alpha$. This implies that $E=\bigcup_{\alpha \in \text{acc } D^\prime } C_\alpha $ is a thread in the ground model.  
