You have five tasks: T1, T2, T3, T4, and T5. If T1 completes, T2 and T3 should be triggered in parallel. If T2 and T3 complete successfully, then T4 and T5 should be triggered in parallel.

The dependency graph can be represented as a directed acyclic graph (DAG). T1 is the root; T2 and T3 depend on T1; T4 and T5 depend on both T2 and T3. A simple solution is to use a thread pool or async framework. First, run T1. When it finishes, submit T2 and T3 to the pool. Use a latch or future join to wait for both to finish. Once both succeed, submit T4 and T5. Error handling: if any task fails, propagate the failure and abort downstream tasks. Complexity: O(n) where n is the number of tasks; parallelism is limited by the number of available workers. This approach scales to larger DAGs and can be implemented with libraries like Java CompletableFuture, Python asyncio, or a workflow engine such as Airflow.