Bronchitis and pneumonia are the common respiratory diseases, of which pneumonia is the leading cause of mortality in pediatric patients worldwide and impose intense pressure on health care systems. This study aims to classify bronchitis and pneumonia in children by analyzing cough sounds. We propose a Classification Framework based on Cough Sounds (CFCS) to identify bronchitis and pneumonia in children. Our dataset includes cough sounds from 173 outpatients at the West China Second University Hospital, Sichuan University, Chengdu, China. We adopt aggregation operation to obtain patients' disease features because some cough chunks carry the disease information while others do not. In the stage of classification in our framework, we adopt Support Vector Machine (SVM) to classify the diseases due to the small scale of our dataset. Furthermore, we apply data augmentation to our dataset to enlarge the number of samples and then adopt Long Short-Term Memory Network (LSTM) to classify. After 45 random tests on RAW dataset, SVM achieves the best classification accuracy of 86.04% and standard deviation of 4.7%. The precision of bronchitis and pneumonia is 93.75% and 87.5%, and their recall is 88.24% and 93.33%. The AUC of SVM and LSTM classification models on the dataset with pitch-shifting data augmentation reach 0.92 and 0.93, respectively. Extensive experimental results show that CFCS can effectively classify children into bronchitis and pneumonia.