Finite strictly decreasing sequences of naturals form a tree with every finite level nonempty but no infinite branch

Let $T$ consist of the empty sequence and all finite strictly decreasing sequences of natural numbers. It is prefix closed. For every $n$, the sequence $(n-1,n-2,\ldots,0)$ is a node of length $n$, so every finite level is nonempty.

The root has infinitely many successors, so $T$ is not finitely branching. An infinite branch would be an infinite strictly decreasing sequence of naturals, but its range would have a least element by The well-ordering principle, after which the branch would have to contain a smaller one. Thus no infinite branch exists, and the missing hypothesis relative to [L1] is exactly finite branching.