# Complexity of reachability in directed rooted forests

I'm trying to figure out the complexity of the reachability problem having as input a directed rooted forest, i.e., given a set of directed rooted trees and two vertices $$s$$ and $$t$$, tell if $$s$$ and $$t$$ are connected by a directed path belonging to one of such trees.

I know the problem is in $$NL$$, since STCON is $$NL$$-complete (and a directed rooted forest is just a kind of directed graph), but I can't manage to find a good lowerbound.

Any hint?

It’s easier to think about it when the edges are written backwards. That is, I will consider the problem formulated as follows: given a directed acyclic graph such that every node has out-degree at most $$1$$, and vertices $$s$$ and $$t$$, determine if $$t$$ is reachable from $$s$$.
To see that it is in L, just follow the unique path starting from $$s$$ until you reach either $$t$$, or a sink.
To see that it is L-hard, use a variant of the usual reduction showing L-hardness of undirected $$s$$-$$t$$-connectivity. Fix a deterministic logspace TM $$M$$; you may assume that it is endowed with a polynomially bounded clock, and whenever it is about to accept, it erases all work tapes and goes into a loop until exhausting the clock. Given an input $$x$$, consider the directed graph consisting of configurations of $$M$$, with edges given by the successor configuration relation, $$s$$ being the initial configuration, and $$t$$ being the accepting configuration (which is unique by our requirements on $$M$$). Since the TM is deterministic, each node has out-degree at most $$1$$, and since the TM is clocked, the graph is acyclic.