Decision tree learning can be roughly classified into two categories: static and incremental inductions. Static tree induction applies greedy search in splitting test for obtaining a global optimal model. Incremental tree induction constructs a decision model by analyzing data in short segments; during each segment a local optimal tree structure is formed. Very Fast Decision Tree [4] is a typical incremental tree induction based on the principle of Hoeffding bound for node-splitting test. But it does not work well under noisy data. In this paper, we propose a new incremental tree induction model called incrementally Optimized Very Fast Decision Tree (iOVFDT), which uses a multi-objective incremental optimization method. iOVFDT also integrates four classifiers at the leaf levels. The proposed incremental tree induction model is tested with a large volume of data streams contaminated with noise. Under such noisy data, we investigate how iOVFDT that represents incremental induction method working with local optimums compares to C4.5 which loads the whole dataset for building a globally optimal decision tree. Our experiment results show that iOVFDT is able to achieve similar though slightly lower accuracy, but the decision tree size and induction time are much smaller than that of C4.5.