Subtree Counts and the Maximum Mean Subtree Order

For a finite simple graph $G$ of positive order, let $s_k(G)$ be the number of its $k$-vertex tree subgraphs. We prove that, among graphs of a fixed order $n$, the ratios $s_k(G)\mathrm{/}{s}_k(K_n)$ form a nonincreasing sequence in $k$. It follows that the complete graph maximizes the mean subtree order: $\mu (G)\le \mu (K_n)$, with equality if and only if $G$ is complete. This monotonicity theorem also establishes the $s_{n-1}$-to-$s_n$ ratio conjecture posed in recent work on extremal mean subtree order. The proof lifts a known coupling of rooted Cayley-tree shapes of consecutive orders to uniform labeled tree subgraphs of $K_n$ that are nested at every sample point. Event inclusion gives the count inequalities, and a double-sum identity gives the mean bound.