The nth small Schröder number is s(n) = Σ_k≥0 s(n, k), where s(n, k) denotes the number of plane rooted trees with n leaves and k internal nodes, each having at least two children. In this manuscript, we focus on the weighted small Schröder numbers s_d(n) = Σ_k≥0 s(n,k)d^k, where d is an arbitrary fixed real number. We provide recursive and asymptotic formulas for s_d(n), as well as some identities and combinatorial interpretations for these numbers. We also establish connections between s_d(n) and several families of Dyck paths.