Generalization of free magmas for nested structures

Question

Consider a nonempty set $X$. What is the name / concept that gives rise to (the set of) all $X$ labeled planar trees e.g.

      *
    /   \
   /      *
  *     /   \
 / \   x3    *
x1  x2     / | \
         x4  x5 x6

which could alternatively be described via a nested list $$\Big[ \big[x_1, x_2 \big], \big[x_3, [x_4, x_5, x_6]\big] \Big]$$

or say

      ___ *___
    /     |    \
   /      x3     \
  *             __*__
 / \          /      *
x1  x2       x4    / | \
                 x5  x6 x7

which gives rise to the bracketing $$\Big[\big[x_1, x_2\big], x_3, \big[x_4, [ x_5, x_6, x_7]\big]\Big]$$

One way that would almost make sense (but doesn't capture it) is something like the free non-associative magma on the free monoid (or semigroup) on $X$ (if we want to avoid empty leaves.) The free monoid on $X$ would provide the ``associative bits'' i.e. the leaves i.e. the (arbitrary size) arrays $[x, y, z, \dots ]$ and the free magma on those provides all the non associative ways to concatenate them. The problem is only with the fact that the free magma is constructed recursively from pairs of elements in lower degrees. Here I allow $k\geq 1$ tuples. In other words I want to use the recursion

$$X_1 = X, ~X_p = \coprod_{\substack{n_1 + \cdots +n_r = p \\ 1\leq n_i, r\geq 1}} X_{n_1}\times\cdots\times X_{n_r}$$

So that for instance

*
|
*
|
x

(corresponding to $[[x]]$) is allowed

Such structures appear in computer science e.g. in RLP encoding which provides a way to encode nested structures. It seems to me that they also describe the (set of) $n$-ary operations of a free operad generated by one $n$-ary operations $\mu_n$ for every $n \geq 1$. I remember that (some form of) rooted planar trees are of importance in describing operations. I don't know anything about operads beyond simple definitions. I'm more interested in the computer-science-math description of these tings.

Edit. Maybe it can be described as the colimit of a system of sets defined as $X_1 = X$ and $X_{n + 1} = F(X_1 \sqcup X_2 \sqcup \cdots \sqcup X_n)$ where $F$ is the free semigroup functor?

Edit 2 Or maybe like so: set $GX = X \sqcup FX$. There is a natural transformation $1_\mathrm{Set} \to G$ by inclusion in the first factor. Then define the set which this question is about as the colimit of the diagram

$$ X \to GX \to GGX \to \cdots$$

Edit 3 the above doesn't work, it will generate $[x]$ twice ...