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With the growing commercial interest in blockchains, permissioned implementations have received increasing attention. Unfortunately, the BFT consensus algorithms that are the backbone of most of these blockchains scale poorly and offer limited throughput. Many state-of-the-art algorithms require a single leader process to receive and validate votes from a quorum of processes and then broadcast the result, which is inherently non-scalable. Recent approaches avoid this bottleneck by using dissemination/aggregation trees to propagate values and collect and validate votes. However, the use of trees increases the round latency, which ultimately limits the throughput for deeper trees. In this paper we propose Kauri, a BFT communication abstraction that can sustain high throughput as the system size grows, leveraging a novel pipelining technique to perform scalable dissemination and aggregation on trees. Our evaluation shows that Kauri outperforms the throughput of state-of-the-art permissioned blockchain protocols, such as HotStuff, by up to 28x. Interestingly, in many scenarios, the parallelization provided by Kauri can also decrease the latency.