There sorta kind of is and isn't! But we need to refine our terminology to understand what's going on.
We have a hierarchy like this in graph theory, which flows from general at the top to more specific/restricted at the bottom:
Graph: a collection of nodes (vertex) and paths (connections between vertex)
Connected graph: a collection of nodes where you can pick any one start node and reach any other end node. The choice of start and end is arbitrary.
Tree: A connected graph where there is one and only one unique path between any one start node and any one end node. There are no alternate paths which do not backtrack (visit a node twice).
Rooted Tree: Just as a tree, but there is one specific item picked as being the root. From these we also define leaves, which is any node that must be an end-node when visited from the root - without visiting a node twice once you hit a leaf you cannot go anywhere, and the path is terminated. Leaves can also be called a "terminal vertex".
Note that even a rooted tree does not necessarily have direction - you are free to start at a leaf and proceed to the root, etc.
If a tree is implemented as a single-linked data structure, this can be thought of as a directed tree if there is no back-link available - once you leave the root for one of the connected nodes, you can't get back!
But such an implementation of a tree is not necessary in the definition of a tree! One can implement a tree with double-linked lists, such that you are free to do things like move towards the root (often referred to as a Parent) or away from the root (Child).
So if you want a data structure where you can start at a leaf and work to the root, you just need one that supports a "visit parent" function. You can also trivially define a system which does not permit moving from root (Parent) to Child, only from Child to Parent, and this sounds like what you are looking for. You basically just remove "visit child" functionality.
However, I am not aware of any formal name for such a structure, or what use such a structure would have. Trees that start from a root are useful, and ones that allow walking up and down are useful, but ones that only allow one-way movement towards the root is of such specialized/limited usefulness that I doubt it has a special name reserved for it. But I suppose anything is possible!
As a practical matter, you can easily implement one from existing directed trees or directed rooted trees.