Throughout this question, I shall let $A^{\mathcal{U}}$ denote the ultrapower of a structure $A$ by an ultrafilter $\mathcal{U}$. Suppose that $T$ is an Aronszajn tree and $\mathcal{U}$ is an ultrafilter on a countable set $I$. Let $T_{\alpha}$ denote the $\alpha$-th level of a tree $T$. Let $T^{(\mathcal{U})}$ denote the tree where the $\alpha$-th level of $T^{(\mathcal{U})}$ is the ultrapower $(T_{\alpha})^{\mathcal{U}}$ and where if $(x_{i})_{i\in I}/\mathcal{U}\in(T_{\alpha})^{\mathcal{U}},(y_{i})_{i\in I}/\mathcal{U}\in (T_{\beta})^{\mathcal{U}}$, then $(x_{i})_{i\in I}/\mathcal{U}<(y_{i})_{i\in I}/\mathcal{U}$ iff $\{i\in I|x_{i}<y_{i}\}\in\mathcal{U}$. Then $T^{(\mathcal{U})}$ is clearly a tree. In other words, we trim the ultrapower $T^{\mathcal{U}}$ so that it becomes a tree $T^{(\mathcal{U})}$.

Does the tree $T^{(\mathcal{U})}$ have a branch of length $\omega_{1}$? What about when we assume that $T$ is a Suslin tree? What happens when we consider ultraproducts of countably many Aronszajn trees instead of ultrapowers?