= Solution
A <set-theoretic tree> is a partial order for which the strict predecessors of every node are well-ordered by the tree order. The height of a node is that predecessor <order type>, and $T_\alpha$ is the set of nodes of height $\alpha$. Under the ordinary height-and-width definition, a <kappa-tree> satisfies
$$
\boxed{\operatorname{ht}(T)=\kappa,\qquad 0<|T_\alpha|<\kappa\quad(\alpha<\kappa).}
$$
For an arbitrary <cardinal> in this definition one must distinguish it from additional conventions such as being well-pruned, normal or splitting. Splitting means that every node has two incompatible extensions; it is not implied by the height-and-width clauses. This distinction is material in Question 5(ii)(a).
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