Let $T$ be a single directed tree, by parameters $(\kappa, \lambda, \zeta)$ of $T$ we mean: the number of root nodes in $T$, the strict upper bound on the number of children nodes per a node in $T$, the strict upper bound on the level of a node in $T$ respectively.



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> Would ZFC be interpreted in a Graph theory that stipulates the existence of any directed tree with parameters $(1, \operatorname{icc}, \omega)$, where $\operatorname {icc}$ stands for the first weakly inaccessible cardinal.
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By a Graph theory I mean a first order theory about graphs, i.e. a theory that extends FOL+Equality with the primitive notions of *node* and *arrow* and set *membership*, and axiomatize Extensionality over sets, and unrestricted construction of sets of nodes and arrows, and restricts sets to only those of nodes and arrows. For simplicity a node cannot be an arrow. Of course by an arrow here it is mean a "directional" edge, and for any two nodes there exists an arrow stemming from one of them to the other in each direction, also no arrow can take the role of a node, so no arrow can link arrows or an arrow and a node.



Can ZFC sets be interpreted as single rooted trees with accessible degree and countable height?
The details of your graph theory will matter - how, for example, are you going to experess "$(1,\mathsf{icc}, \omega)$" in your setting? - but certainly some version of this will work: in a (well-founded) model $M$ of $\mathsf{ZFC}$ there's a natural way to code sets by well-founded trees, and so a "rich enough" theory of trees will interpret $\mathsf{ZFC}$.


This coding idea (if not the specific interpretability fact you're asking about) actually gets used from time to time in purely technical ways, e.g. the coding of elements of $L\_{\omega\_1^L}$ by reals definably in $L\_{\omega\_1^L}$ (see the beginning of [Sacks' *Higher recursion theory*](https://projecteuclid.org/ebooks/perspectives-in-logic/Higher-Recursion-Theory/toc/pl/1235422631)).


