Which notions of forcing add a cofinal branch to an $\omega_1$-tree? I'd like to know more about forcing to add a cofinal branch to an $\omega_1$-tree.  
Question 1:  
What kinds of forcings add cofinal branches to $\omega_1$-trees?  What kinds of forcings cannot?
The answer may depend on the type of tree.  I don't expect a full characterization of such forcings, but would like to know what experience anyone has with forcing branches.  
Question 2:  
Is the following statement correct?  If $\mathbb{P}$ is a notion of forcing which has the countable chain condition and $\mathbb{P} \times \mathbb{P}$ also has the countable chain condition, then $\mathbb{P}$ cannot force a cofinal branch in an $\omega_1$-tree.
If the statement is correct, what is the proof?  A non-example is that a Souslin poset is ccc, but the product of two Souslin posets is $\it{not}$ ccc (since the set of pairs of immediate sucessors of each node in the tree gives an uncountable antichain) and a Souslin poset $\it{can}$ add a cofinal branch to a Souslin tree.  
Question 3: 
Sometimes (as in the case of a special Aronszajn tree) a notion of forcing which satisfies the countable chain condition cannot add a branch to an $\omega_1$-tree because any forcing which adds a cofinal branch to the tree will collapse $\omega_1$ and ccc posets preserve $\omega_1$. 
Does anyone have any experience with an $\omega_1$-tree such that no ccc forcing can add a branch, but the reason is not because adding a cofinal branch will collapse $\omega_1$?
These questions are related to and arose because of Joel Hamkins' open question on mathoverflow: 
Can there be an almost-special not-fully-special Aronszajn tree?
 A: Tanmay Inamdar has pointed out to me that Lemma 2.8 of my paper with Stevo Todorcevic, ``Chain conditions in maximal models", is related to Question 3 here, and also to Joel and Arthur's questions. In this Lemma, we assume that $S$ is a stationary, co-stationary subset of $\omega_{1}$, and that $\Diamond$ holds on the compliment of $S$. By a relatively straightforward construction, we build an $\omega_{1}$-tree $T$ and a set $A \subseteq T$ such that if $P$ is any path through $T$ intersecting $A$ at uncountably many levels (as any generic branch will do), then this set of levels is a club disjoint from $S$. Furthermore, forcing with $T$ does not collapse $\omega_{1}$. This doesn't quite answer the questions above, as it is still possible that some c.c.c. forcing adds an uncountable branch to the tree formed by removing the nodes in $A$ from $T$. However, it seems that one can modify the construction to specialize $T\setminus A$, that is, to add a function $f \colon T \setminus A \to \omega$ such that whenever $p, q \in T \setminus A$ are compatible, and no member of $A$ lies in between $p$ and $q$, then $f$ is injective on the interval from $p$ to $q$. Then any outer model with an uncountable branch must either collapse $\omega_{1}$ or make $S$ nonstationary. 
A: An alternate argument is simply to let $B$ be a name for a new branch and choose for each $\alpha \in \omega_1$ conditions $p_\alpha$ and $q_\alpha$ and $t_\alpha$ in the $\alpha^\text{th}$ level of the tree
such that
 $p_\alpha $ forces that $t_{p,\alpha}\in B$ and $q_\alpha $ forces that $t_{q,\alpha}\in B$ and
$t\leq t_{q,\alpha}$ and $t\leq t_{p,\alpha}$.
(This is possible because otherwise $B$ is not new.) Now it follows that $\{(p_\alpha, q_\alpha)\}_{\alpha\in\omega_1}$ must be an antichain in $\mathbb{P}\times\mathbb{P}$ (or at least can be thinned out to an uncountable antichain) --- this is the same argument as that the product of trees is never ccc.
A: The answer to question 2 is yes, and I believe that this is due to
Silver. (It may be in Jech's book.) The hypothesis that
$\mathbb{P}\times\mathbb{P}$ is c.c.c. is equivalent to the
assertion that $\mathbb{P}$ remains c.c.c. after forcing with
$\mathbb{P}$.
Suppose now that $\mathbb{P}$ adds a new branch $t$ through a
ground model tree $T$ of height $\omega_1$, and we don't even need
it to have countable levels. Let $\dot t$ be a $\mathbb{P}$-name
for this new branch, and let's assume $\Vdash\dot t$ is a new
branch through $\check T$. Since $\mathbb{P}$ is c.c.c., $\omega_1$
is preserved. Consider the Boolean values $b_\alpha=\[\[\check
t_\alpha\subset\dot t\]\]$, for $\alpha<\omega_1$, where
$t_\alpha=t\upharpoonright\alpha$ is the new branch up to $\alpha$
(and note that for any fixed $\alpha$, this is an element of the
ground model). If $\alpha<\beta$, then $b_\beta\leq b_\alpha$, and
furthermore, these Boolean values must eventually descend, since
otherwise we would be able to define $t$ in the ground model. So in
the extension, we may refine this to a strictly descending
$\omega_1$-sequence of Boolean values. The differences between
these values makes an uncountable antichain in $\mathbb{P}$ in the
extension, contrary to the assumption that
$\mathbb{P}\times\mathbb{P}$ is c.c.c.
