Directed acyclic graphs are commonly used to model scientific workflows, by expressing dependencies between tasks, as well as the resource requirements of the workflow. As a special case, rooted directed trees occur in several applications, for instance in sparse matrix computations. Since typical workflows are modeled by large trees, it is crucial to schedule them efficiently, so that their execution time (or makespan) is minimized. Furthermore, it is usually beneficial to distribute the execution on several compute nodes, hence increasing the available memory, and allowing us to parallelize parts of the execution. To exploit the heterogeneity of modern clusters in this context, we investigate the partitioning and mapping of tree‐shaped workflows on two types of target architecture models: in AM1, each processor can have a different memory size, and in AM2, each processor can also have a different speed (in addition to a different memory size). We design a three‐step heuristic for AM1, which adapts and extends previous work for homogeneous clusters [Gou C, Benoit A, Marchal L. Partitioning tree‐shaped task graphs for distributed platforms with limited memory. IEEE Trans Parallel Dist Syst 2020; 31(7): 1533–1544]. The changes we propose concern the assignment to processors (accounting for the different memory sizes) and the availability of suitable processors when splitting or merging subtrees. For AM2, we extend the heuristic for AM1 with a two‐phase local search approach. Phase A is a swap‐based hill climber, while (the optional) Phase B is inspired by iterated local search. We evaluate our heuristics for AM1 and AM2 with extensive simulations, and we demonstrate that exploiting the heterogeneity in the cluster significantly reduces the makespan, compared to the state of the art for homogeneous processors.