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We study a link between complete non-ambiguous trees (CNATs) and permutationsexhibited by Daniel Chen and Sebastian Ohlig in recent work. In this, theyassociate a certain permutation to the leaves of a CNAT, and show that thenumber of $n$-permutations that are associated with exactly one CNAT is$2^{n-2}$. We connect this to work by the first author and co-authors linkingcomplete non-ambiguous trees and the Abelian sandpile model. This allows us toprove a number of conjectures by Chen and Ohlig on the number of$n$-permutations that are associated with exactly $k$ CNATs for various $k >1$, via bijective correspondences between such permutations. We also exhibit anew bijection between $(n-1)$-permutations and CNATs whose permutation is thedecreasing permutation $n(n-1)\cdots1$. This bijection maps the left-to-rightminima of the permutation to dots on the bottom row of the corresponding CNAT,and descents of the permutation to empty rows of the CNAT.

Complete non-ambiguous trees and associated permutations: connections through the Abelian sandpile model