What should I call a poset with the property that each element has a unique predecessor?

(I'm actually interested in the special case in which there are no infinite descending chains.)

CLARIFICATION: I mean each element has a unique immediate predecessor.
As you go higher up there can be branching, but never collapsing. That is, I don't want to allow $a < z$ and $b < z$ with $a$ and $b$ incomparable; but I am happy with $a< y$ and $a< z$ with $y$ and $z$ incomparable.

What should I call a poset with the property that each element has AT MOST ONE predecessor?

(I'm actually interested in the special case in which there are no infinite descending chains.)

CLARIFICATION: I mean each element has a unique immediate predecessor.
As you go higher up there can be branching, but never collapsing. That is, I don't want to allow $a < z$ and $b < z$ with $a$ and $b$ incomparable; but I am happy with $a< y$ and $a< z$ with $y$ and $z$ incomparable.

FURTHER:

1.
The immediate predecessor should be the largest predecessor. If I write $i-1$ for the unique immediate predecessor of $i$, then $i-1 \leq x \leq i$ forces either $x = i-1$ or $x = i$.

2. I should have said -- and now do say -- that each element has at most one immediate predecessor.

What should I call a poset with the property that each element has a unique predecessor?

(I'm actually interested in the special case in which there are no infinite descending chains.)

What should I call a poset with the property that each element has a unique predecessor?

(I'm actually interested in the special case in which there are no infinite descending chains.)

CLARIFICATION: I mean each element has a unique immediate predecessor.
As you go higher up there can be branching, but never collapsing. That is, I don't want to allow $a < z$ and $b < z$ with $a$ and $b$ incomparable; but I am happy with $a< y$ and $a< z$ with $y$ and $z$ incomparable.