Have these (extremely simple) classes of algebraic structures been considered in the literature? If so, what are they called? 
Questions. Have the following kinds algebraic structures been considered in the abstract algebra literature etc.? If so, what are they really called? (I have used made-up terminology for the sake of the question.)



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*A woodland is a set equipped with an $n$-ary operation $f_n$ for each integer $n \geq 0$. (Observe that every woodland has a distinguished element corresponding to the case $n=0$).

*A jungle is a woodland such that we can permute the arguments of $f_n$ willy-nilly. In particular, $(X,f_*)$ is a jungle iff for all integers $n \geq 0$ and all permutations $\pi$ of $\{0,\ldots,n-1\},$ it holds that $f_n(x_0,\ldots,x_{n-1}) = f_n(x_{\pi(0)},\ldots,x_{\pi(n-1)}).$

Motivation. The rooted trees of graph theory form a jungle in an obvious way, and this is (isomorphic to) the initial jungle. Similarly, the ordered rooted trees form a woodland in an obvious way; and, this is the initial woodland.

 A: A "woodland" would be an algebra over the free nonsymmetric operad $W$ with one generating operation in each arity $n \ge 0$ and no relations. Similarly a "jungle" would be an algebra over the free symmetric operad $J$ with one generating operation in each arity $n \ge 0$ and $\Sigma_n$ acts trivially on the $n$th generating operation. Trees appear naturally because this is what operads are built on. It seems to me that the set of (resp. ordered) rooted trees is in fact the free $W$-algebra (resp. $J$-algebra) over the empty set (the initial set), which explains why they're initial in their category of algebras.
I don't believe these things have been studied on their own, as they have very little structure; consider that a magma is a woodland where all the $f_n$ are trivial for $n \neq 2$, and magmas aren't particularly studied. They're more of an intermediary technical tool.
One (rather silly) thing I can say is that if you consider a linear version of this and remove the operations in arity $0$ and $1$, then the first one is the ungraded underlying operad of the dg-operad $A_\infty$ controlling homotopy associative algebras (it would have $\operatorname{deg} f_n = n - 2$ and a differential $\partial(f_n)$ that I don't want to write down). I guess you can relate the second one to some shifted version of $L_\infty$. I doubt this is what you had in mind though.
