This randomized phase II/III trial studies how well giving combination chemotherapy with or without rituximab works in treating younger patients with stage III or stage IV non-Hodgkin lymphoma or B-cell acute leukemia. Drugs used in chemotherapy work in different ways to stop the growth of cancer cells, either by killing the cells or by stopping them from dividing. Monoclonal antibody, such as rituximab, can block cancer cells growth in different ways. Some block the ability of cancer cells to grow and spread. Others find cancer cells and help kill them or carry cancer-killing substances to them. It is not yet known whether giving combination chemotherapy together with rituximab is more effective in treating patients with non-Hodgkin lymphoma or B-cell acute leukemia.