In this paper, we present and study the weak single-crossover domain on trees, which is a generalization of the well-studied single-crossover domain in social choice theory. We design a polynomial-time algorithm to recognize preference profiles belonging to this domain, and develop an efficient information gathering algorithm that works even when the preferences can be accessed only sequentially and the underlying single-crossover tree structure is not known in advance. We also prove a consistent lower bound on the query complexity of the information gathering algorithm when the number of voters is much larger than the number of candidates. We prove a lower bound of $\Omega(m^2\log n)$ on the number of queries that an algorithm must make to gather a single-crossover profile when random queries are allowed. This solves the open problem of the previous paper and proves the optimality of the preference information gathering algorithm when random queries are allowed.