Normal and splitting trees

For a tree as in Set-theoretic trees, heights, levels, branches and antichains, call $T$ normal when it has exactly one root, every $t\in T$ has an extension in $T_{\beta }$ whenever $\mathrm{ht}(t)<\beta <\mathrm{ht}(T)$, and distinct nodes on the same nonzero limit level have distinct strict predecessor sets.

A node $u$ is an immediate successor of $t$ if $t{<}_{T}u$ and there is no $v$ with $t{<}_{T}v{<}_{T}u$. Call $T$ splitting when every node $t$ has at least two distinct immediate successors whenever $\mathrm{ht}(t)+1<\mathrm{ht}(T)$. Splitting is an additional condition, not part of normality here. No cardinal bound on levels is included in either adjective.

The empty tree is not normal. A singleton tree is normal and vacuously splitting. Nodes on a last level have no splitting requirement; limit-level uniqueness is required only at nonzero limits. These conventions separate conditions that Monk bundles into his normal-tree terminology.