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We classify connected spanning convex subgraphs of the square cycles. We thenshow that every spanning tree of $C_n^2$ is contained in a unique nontrivialconnected spanning convex subgraph of $C_n^2$. As a result, we obtain a purelycombinatorial derivation of the formula for the number of spanning trees of thesquare cycles.

Convex subgraphs and spanning trees of the square cycles