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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 $T$?

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  • $\begingroup$ You’re not missing anything. Unique limits can always be introduced without changing the height or cardinality, but the arity is another matter altogether. If I recall correctly, the usual construction of an $\omega_1$-Aronszajn tree produces an $\omega$-ary $\omega_1$-tree. $\endgroup$ – Brian M. Scott Mar 27 '15 at 5:59
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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 $\omega$.

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  • $\begingroup$ Welcome to MSE. Could you please expand on your example, to make this a fleshed-out answer? $\endgroup$ – A.P. Mar 27 '15 at 16:24

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