Very low birth weight premature infants often suffer from a variety of respiratory problems including respiratory distress syndrome (RDS), hypopnea and periodic breathing, and apnea. These conditions are likely related to immaturity of the respiratory centers, yet how respiratory rhythms originating from these centers, including their complexity, relate to demographic measures of prematurity remains largely unknown. In 39 mild RDS neonates (22 males, gestational age = 28±2 weeks; weight = 1036±234 g), we derived the univariate association between complexity of two respiratory rhythms [respiratory rate (RR) and tidal volume (VT)] to postmenstrual age, gestational age, postnatal age and weight at time-of-study. RR and VT rhythm complexities were assessed using approximate entropy, sample entropy, base scale entropy, and forbidden words entropy estimated for 300 consecutive breaths determined from respiratory inductance plethysmography - irrespective of breathing effort rate or amplitude - collected during sleep while on nasal continuous positive airway pressure (4-6 cmH2O). Both RR and VT exhibited increased complexity with increased maturity, but only in terms of base scale entropy and forbidden words entropy which are based on pattern matching rather than approximate and sample entropies, which are based on conditional probabilities. Specifically, RR complexity as measured by forbidden word entropy increased with increasing weight (r=.502), postconceptional age (r=.423), and gestational age (r=493). As measured by base scale entropy, it increased with increasing weight (r=.488) and postconceptional age (r=.390). VT complexity, measured by base scale entropy was greater with increased postnatal age (r=.428). Our results indicate that respiratory rhythms become more complex with increasing levels of maturity as indicated by increased weight and several age parameters. This reiterates the importance of the latter weeks of gestation on the maturation process of respiratory centers in the brain, and suggests a promising use of entropy measures in exploring respiratory maturation in infants.