It comes from set theory. To see that these ideas were around before computers and that they were explicitly called trees consider the following famous tree. The convention set theorist sometimes use (some use use the convention you gave) is the opposite (the root $\hat{0}$ is the smallest, i.e. $\hat{0}<x$ for all $x\neq \hat{0}$, not the biggest ... this is keeping in line with the equivalence of the principle of induction and the well-ordering principle and is more natural ... i.e. a "zero-like" object is the root ... I will keep that convention so just replace all of the $<$ with $>$ in the proceeding discussion) The idea is the following:

Theorem From any partial (well)-order we can construct a tree and from any tree we can construct a partial (well)-order.

Remark Partial (well)-order is a name I made up so as to artificially distinguish between trees and the relation I am about to define:

Let $P = (V, E) =(V, \leq_E ) $ be a partial (well)-order then it satisfies the following

1. $(\forall x) \ x \leq x$
2. $(\forall x,y) \ x \leq y \implies y \leq x $
3. $(\forall x,y) \ x \leq y \land y \leq x \implies x = y $
4. $(\exists \hat{0} )(\forall x) \ \hat{0} \leq x$
5. $(\forall x) \ (\exists 0_{x} ) \ y \in \{ t \in V \ | \ t \leq x\} \implies 0_{x} \leq y $ ,

The first 3 state that $\leq$ is a partial order and the last states that the predecessor sets are well ordered.

(Proof of Partial implies Tree) Let $P = (V, \leq_E ) $ be a well partial order then connect $v$ to $w$ if $w$ is the least element such that $v \leq w$, i.e. $ w = \min \{x \in V \ | \ v \leq x\}$ or $w$ covers v. It is easy to see that there are no cycles (because of rule 3) and that the graph is connected (there is always a path from $\hat{0}$ to any $v$ by induction because $\leq$ is a well order)

(Proof of Tree implies Partial) Let $x \leq y$ if there exists a path from $x$ to $y$, it is tedious but easy to see that all of the rules above are satisfied.

You may thinking, "man I didn't want an explanation involving set theory"

Let me help you dispel those heretical thoughts with the following observations

Most of the famous early famous computer scientists got most of their techniques from set theory

1. The solution to the halting problem is an adaptation of a very beautiful proof by Georg Cantor that there are different sizes of infinity
2. Similarly so is the Entscheidungsproblem
3. Similarly, the proof that exponential time does not equal polynomial time
4. Von Neumann (Von-Neumann architecture) also came up with the Von-Neumann Universe of sets

This should come as no surprise set theory is concerned with the ontological status of the mathematical universe and the idea that all of the objects of our mathematical experience can be constructed via recursion ... and isn't the big idea behind every algorithm/computer program ... some recursive construction ... set theory is like a programming language for the platonic realm.

I hope that I have whet your curiosity towards the beauty of sets. Good luck on your mathematical voyages! Bon Voyage!