= Solution
A <tree with unique limits> has no distinct nodes at a limit level with the same history. Precisely, for a <limit ordinal> $\delta$ and $s,t\in T_\delta$,
$$
\boxed{\bigl(\forall\beta<\delta\quad s\upharpoonright\beta=t\upharpoonright\beta\bigr)\Longrightarrow s=t.}
$$
Here $s\upharpoonright\beta$ denotes the unique predecessor of $s$ of height $\beta$. This is uniqueness of a limit node when it exists, not a requirement that every cofinal <chain in a partial order> below a limit level acquire a limit node.
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