What’s a tree? First define a sequence or length n> 0 as a total map s: {1,.. n}→ X. Let Nil be the empty sequence. If s is a sequence of length n, and x is an element, then u = sx is the sequence of length n+1 so that u(n)=x and for i < n, u(i)=s(i). That is sx is the sequence obtained by appending element x to sequence s on the right. If X is a set, let X* be the set of finite sequences over X, the minimal set that includes Nil and sx so that if s is an element of X*, sx is also an element. If s is a sequence, then either s=Nil or s = ux for some sequence u that has a length one less than the length of s. Define the predecessor of a sequence by Pred(Nil)=Nil and Pred(ux)=u.
A tree is a map T: X* → Y where whenever T(s) is defined, T(Pred(s)) must be defined. Note that T(Nil) must be defined if T is defined on any elements of X* at all.
Let V the set of vertices contain exactly those elements s of X* so that T(s) is defined. If V is not empty, Nil is in V and call Nil the “root”. Let E the set of labeled edges consist of exactly all triples (s,x,sx) where T(sx) is defined. Then (V,E) is the standard graph theoretic definition of a tree. Suppose, on the contrary that (V,E) is a tree with labeled edges, that is it is a undirected graph that is connected but has no cycles. Then define T(Nil) = root and T(sx) = v s.t. there is an edge labeled x from T(s) to v.
So a sequence is a map from numbers to elements and a tree is a map from sequences to elements.