I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup whose binary relation is the join operation. Topologically, I think of them with an Alexandrov topology, a topology whose open sets are just the lower sets.
I'm interested in these two conceptions of a tree--one algebraic and combinatorial and one topological, concerned with all of the possible "spaces" in a tree. You can even think of the open sets as continua, like an interval on the real line, right?
My main question is this: is there any way to connect these two views in something like algebraic topology, where there's a functor from Top to Grp? Is there some way to imagine a functor from the category of topological spaces to the category of posets? After all, every tree is a poset (and semigroup) and has an associated topological space. Please excuse my relative lack of knowledge--I'm self-taught in mathematics and doing a PhD in a humanities field.