Single vs Double Umbilical Cord Blood Transplants in Children With High Risk Leukemia and Myelodysplasia (BMT CTN 0501)

Inclusion Criteria: * Two partially HLA-matched UCB units. Units must be HLA-matched minimally at 4 of 6 HLA-A and B (at intermediate resolution by molecular typing) and DRB1 (at high resolution by molecular typing) loci with the patient, and the units must be HLA-matched at 3 of 6 HLA- A, B, DRB1 loci with each other (using same resolution of molecular typing as indicated above). Two appropriately HLA-matched units must be available such that one unit delivers a pre-cryopreserved nucleated cell dose of at least 2.5 x 10\^7 per kilogram and the second unit at least 1.5 x 10\^7 per kilogram. * Acute myelogenous leukemia (AML) at the following stages: 1. High risk first complete remission (CR1), defined as the following: * Having preceding myelodysplasia (MDS) * High risk cytogenetics (high risk cytogenetics: del (5q) -5, -7, abn (3q), t (6;9) complex karyotype \[at least 5 abnormalities\],)the presence of a high FLT3 ITD-AR (\> 0.4) * Requiring more than 1 cycle of chemotherapy to obtain complete remission (CR); * FAB M6 2. Second or greater CR 3. First relapse with less than 25% blasts in bone marrow 4. Morphologic complete remission with incomplete blood count recovery * Therapy-related AML for which prior malignancy has been in remission for at least 12 months * Acute lymphocytic leukemia (ALL) at the following stages: 1. High risk first remission, defined as one of the following conditions: * Philadelphia chromosome-positive adult lymphoblastic leukemia (Ph+ ALL) * Mixed lineage leukemia (MLL) rearrangement with slow early response (defined as having M2 \[5-25% blasts\] or M3 \[more than 25% blasts on bone marrow examination on Day 14 of induction therapy\]) * Hypodiploidy (less than 44 chromosomes or DNA index less than 0.81) * End of induction M3 bone marrow * End of induction M2 with M2-3 at Day 42 * Evidence of minimal residual disease (MRD). If a patient's only high risk criterion is MRD, approval by a protocol chair or protocol officer is required for enrollment. For COG centers, this will only be for MRD greater than 1 percent by flow MRD at the end of extended induction. 2. High risk second remission, defined as one of the following conditions: * Philadelphia chromosome-positive adult lymphoblastic leukemia (Ph+ ALL) * Bone marrow relapse less than 36 months from induction * T-lineage relapse at any time * Very early isolated central nervous system (CNS) relapse (6 months from diagnosis) * Slow reinduction (M2-3 at Day 28) after relapse at any time * Evidence of minimal residual disease (MRD). If a patient's only high risk criterion is MRD, approval by a protocol chair or protocol officer is required for enrollment. For COG centers, this will only be for MRD greater than 1 percent by flow MRD at the end of extended induction. 3. Any third or subsequent CR * NK cell lymphoblastic leukemia in any CR * Biphenotypic or undifferentiated leukemia in any CR or if in first relapse must have less than 25% blasts in bone marrow (BM) * Myelodysplastic syndrome (MDS) at any stage * Chronic myelogenous leukemia (CML) in chronic or accelerated phase * All patients with evidence of CNS leukemia must be treated and be in CNS CR to be eligible for study. * Patients 16 years old or older must have a Karnofsky score of at least 70% and patients younger than 16 years old must have a Lansky score of at least 70%. * Patients with adequate physical function as measured by: 1. Cardiac: Left ventricular ejection fraction greater than 40% or shortening fraction greater than 26% 2. Hepatic: Bilirubin no more than 2.5 mg/dL; alanine aminotransferase (ALT), aspartate aminotransferase (AST), and alkaline phosphatase (ALP) no more than 5 times the upper limit of normal (ULN) 3. Renal: Serum creatinine within normal range for age, or if serum creatinine is outside normal range for age, then renal function (creatinine clearance or GFR) greater than 70 mL/min/1.73 m\^2 4. Pulmonary: Diffusing capacity of the lung for carbon monoxide (DLCO), forced expiratory volume in one second (FEV1), or forced vital capacity (FVC) greater than 50% of predicted value (corrected for hemoglobin); if unable to perform pulmonary function tests, then O2 saturation greater than 92% of room air Exclusion Criteria: * Pregnant (β-positive human chorionic gonadotropin \[HCG\]) or breastfeeding * Evidence of HIV infection or HIV positive serology * Current uncontrolled bacterial, viral, or fungal infection (currently taking medication and progression of clinical symptoms) * Autologous transplant less than 12 months prior to enrollment * Prior autologous transplant for the disease for which the UCB transplant will be performed * Prior allogeneic hematopoietic stem cell transplant * Active malignancy other than the one for which the UCB transplant is being performed within 12 months of enrollment * Inability to receive TBI * Requirement of supplemental oxygen * HLA-matched related donor able to donate