Defining computable functions on arbitrary sets
Effective model theory studies computable structures. The collection of all finite trees is a two-sorted computable structure in which one sort consists of vertices (which can be identified with the natural numbers) and the other sort consists of trees. It has the following relations:


$\operatorname{vertex}(x)$, which is true if $x$ is a vertex.
$\operatorname{tree}(x)$, which is true if $x$ is a tree.
$\operatorname{appears}(x,T)$, which is true if $x$ is a vertex that appears in the tree $T$.
$\operatorname{edge}(x,y,T)$, which is true if $x,y$ are two vertices in $T$ which are connected by an edge.


(We don't really need both $\operatorname{vertex}$ and $\operatorname{tree}$, but we do need to be able to express being in the domain of the structure, i.e., either a vertex or a tree.)

Formally speaking, to complete the definition of the structure we have to specify an encoding of the domain, either as strings or as natural numbers (this is an arbitrary choice which doesn't make much difference). Under any reasonable encoding, all the relations above will be computable, and so we have a computable structure. 

A computable presentation of this structure is an isomorphism which maps this structure to another computable structure. This mapping has to be an isomorphism of structures, but it doesn't have to be computable. A computable structure $\mathcal{A}$ is computably categorical if whenevever $\mathcal{A}$ is isomorphic to a computable structure $\mathcal{B}$ (by a computable presentation), then there is a computable isomorphism between the two structures.

Using this vocabulary, what you might be asking is whether the computable structure of finite trees is computably categorical. (There is actually more freedom in your question, since you might want to use a different relational structure.) Some simple computable structures are computably categorical (for example, the rationals with the order relation, and the natural numbers with the successor function) and some aren't (for example, the natural numbers with the order relation), see Computable structures: presentations matter by Shore.