 Solution

A well-pruned set-theoretic tree of height $\kappa$ has the property that every node $t$ extends to every higher level below $\kappa$: if $\mathrm{ht}(t)<\beta <\kappa$, there is $u$ of height $\beta$ with $t{<}_{T}u$. Equivalently, the heights of extensions of every node are unbounded in $\kappa$, since taking predecessors then gives an extension at any prescribed intermediate level. This is stronger than merely having no terminal nodes. For a kappa-tree we use the usual regular uncountable height cardinal and levels of size less than that cardinal.