On Ternary Trees and Fighting Fish

Fighting fish is a combinatorial configuration introduced by Duchi et al. as a new model of branching surfaces that generalizes directed convex polyominoes. We come up with an alternative construction of fighting fish, using a tree structure built on the so-called stem cells of fighting fish. From this perspective, we establish a bijection between ternary trees and fighting fish with a marked strip of cells, which specializes to a direct bijection between left ternary trees and fighting fish. Using these results, we obtain a combinatorial enumeration of the fighting fish of size $n$ by establishing a $(n+1)$-to-2 bijection with the ternary trees having $n$ nodes. We present some additional enumerative results including that the fighting fish with a marked tail and the horizontally symmetric fighting fish are equinumerous with the ordered pairs of ternary trees having a total of a prescribed number of nodes.