We propose a new procedure for testing whether two networks are edge-correlated through some latent vertex correspondence. The test statistic is based on counting the co-occurrences of signed trees for a family of non-isomorphic trees. When the two networks are Erdős-Rényi random graphs $\mathcal{G}(n,q)$ that are either independent or correlated with correlation coefficient $ρ$, our test runs in $n^{2+o(1)}$ time and succeeds with high probability as $n\to\infty$, provided that $n\min\{q,1-q\} \ge n^{-o(1)}$ and $ρ^2>α\approx 0.338$, where $α$ is Otter's constant so that the number of unlabeled trees with $K$ edges grows as $(1/α)^K$. This significantly improves the prior work in terms of statistical accuracy, running time, and graph sparsity.