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.