Cycle-tree order relation

Question

I am trying to express a cycle-tree structure (https://math.stackexchange.com/a/3835246/427611) as an ordered set.

I want to extend the notion of a cycle-tree structure onto continuous structures as well.

Here is what I got so far:

1. A set $T$ with a partial order $\le$ is a tree if for any element $t$ of $T$ the set $\{ s \in T : s \le t \}$ is totally ordered by $\le$;

2. A tree $T$ is connected if for any two elements $x$ and $y$ of $T$ there is an element $z$ such that $z \le x$ and $z \le y$;

3. A branch of a tree is a maximal totally ordered subset of the tree.

(Discrete, dense, and continuous trees)

4. A partial order $\le$ is compatible with a cyclic order $(x, y, z)$ on a set $C$ if $\le$ is a subset of a cut of $(x, y, z)$.

5. A cycle-tree is a tree $T(\le)$ with a cyclic order $(x, y, z)$ on a subset $C$ of $T$ such that:

   • $\le$ is compatible with $(x, y, z)$ on $C$;
   • if $t \le c$ for some elements $t$ of $T$ and $c$ of $C$, then $t \in C$;

6. A cycle-tree is connected if every branch of $T$ has an element from $C$.

7. A cycle-tree is discrete if both $T$ and $C$ are discrete.

(https://en.wikipedia.org/wiki/Cyclic_order)

Is this definition equivalent to the existing one (via directed graphs) in case of discrete structures?
Would it be a correct generalization onto non-discrete structures?

Are there other definitions of a cycle-tree structure via binary and/or ternary relations?
Since any total order induces a cyclic order and vice versa, is there an elegant way to express a cycle-tree as a pure binary or ternary relation on a set?

Update

A branch of a cycle-tree can be viewed as a cut of a round of a partial cyclic order.
Together with the cycle itself the rounds of all the branches form a ROCO:

https://hal.inria.fr/hal-01360144/document

Obviously, not every ROCO can be "cut" into a cycle-tree.
What are the conditions for a ROCO to represent a cycle-tree structure?