1. A speech recognition system using incremental device-based acoustic model adaptation comprising: a model selector configured to select an acoustic model of a multi-model tree by verifying and categorizing a device key transmitted from a user device; a model manager configured to generate and incrementally adapt the multi-model tree by categorizing voice data based on the device key; and a speech recognizer configured to perform speech recognition based on the selected acoustic model and transmit data of which reliability exceeds a predetermined threshold value to the model manager, wherein the device key represents channel features of the user device, and wherein the model manager comprises: a device-based multi-model tree generating part configured to categorize the voice data based on the device key, extract features therefrom and form the multi-model tree, the multi-model tree being categorized by the device key and comprising hierarchical acoustic models, incremental data and device key trees; a data incrementing part configured to receive data of which the reliability exceeds the predetermined threshold value from the speech recognizer, increment phonetic information and data to the multi-model tree, evaluate a degree of the incremented data of each node, and report a result of the evaluation to a model adaptation generating part when the degree exceeds a predetermined reference value; and the model adaptation generating part configured to incrementally adapt acoustic models corresponding to nodes where newly adapted data from the multi-model tree exceeds the predetermined reference value by using the phonetic information transmitted from the data incrementing part.