From crying to babbling and then to speech, infant's vocal tract goes through anatomic restructuring. In this paper, we propose a non-invasive fast method of using infant cry signals with convolutional neural network (CNN) based age classification to diagnose the abnormality of the vocal tract development as early as 4-month age. We study F0, F1, F2, and spectrograms and relate them to the postnatal development of infant vocalization. A novel CNN based age classification is performed with binary age pairs to discover the pattern and tendency of the vocal tract changes. The effectiveness of this approach is evaluated on Baby2020 with healthy infant cries and Baby Chillanto database with pathological infant cries. The results show that our approach yields 79.20% accuracy for healthy cries, 84.80% for asphyxiated cries, and 91.20% for deaf cries. Our method first reveals that infants' vocal tract develops to a certain level at 4-month age and infants can start controlling the vocal folds to produce discontinuous cry sounds leading to babbling. Early diagnosis of growth abnormality of the vocal tract can help parents keep vigilant and adopt medical treatment or training therapy for their infants as early as possible.