We will show there is a topological space for which presheaves are the same thing as trees. We will further show that there is a sheaf on this topological space which has an important relationship with Baire space. We will then use these connections to show how a definition by transfinite recursion can be thought of as an operation on sheaves, and how the well-definedness of such a definition can be through of as a property of the sheaf we are working on. This will then allow us to expand our notion of definition by transfinite recursion to more general structures than well-founded trees. We will then mention how these techniques can be used to prove a variant of the Suslin-Kleene separation theorem.