Trees are defined as posets $(T,<)$ such that for all $x \in T$ the set of predecessors of $x$ is well ordered by $<$. A $\kappa$-tree has height $\kappa$ and every level $T_{\alpha}$ has cardinality less than $\kappa$.
I've seen references to an "obvious" way of viewing a $\kappa$-tree as a subset of $^{<\kappa}2$ such that $f \in T_{\alpha}$ iff $f:\alpha \rightarrow 2$, and such that a cofinal branch in $T$ is a function $b:\kappa \rightarrow 2$ such that $b \upharpoonright \alpha \in T_{\alpha}$ for each $\alpha < \kappa$.
I believe I am missing something, but I can only see an obvious way to do this if every element of $T$ has (at most) two immediate successors and $T$ has unique limits. Is there a way to view a $\kappa$-tree in this way without these restrictions on $T$?
The unique limit restriction cannot be dropped. For example you can construct a tree whose restriction to finite levels are two branching but it has more than continuum many nodes at level $\omega$.