Data aggregation is an essential operation for the sink to obtain summary information in a wireless sensor network (WSN). The problem of minimum latency aggregation schedule (MLAS) which seeks a fastest and conflict-free aggregation schedule has been well studied when nodes are always awake. However, in duty-cycle WSNs, nodes can only receive data in the active state. In such networks, it is of great importance to exploit the limited active time slots to reduce aggregation latency. Unfortunately, few studies have addressed this issue, and most previous aggregation methods rely on fixed structures which greatly limit the exploitation of the active time slots from neighbors. In this paper, we investigate the MLAS problem in duty-cycle WSNs without considering structures. Two distributed aggregation algorithms are proposed, in which the aggregation tree and a conflict-free schedule are generated simultaneously to make use of the active time slots from all neighbors. Compared with the previous centralized and distributed methods, the aggregation latency and the utilization ratio of available time slots are greatly improved. This paper also proposes several adaptive strategies for handling network topology changes without increasing the aggregation latency. The theoretical analysis and simulation results verify that the proposed algorithms have high performance in terms of latency and communication cost.