I understand that we select a subset of predictors and data points to build each tree in the random forest. The tree is fully grown till the terminal nodes are pure. Can there be a chance that for a given set of predictors and data points, terminal nodes cannot be made pure, within a prespecified tree depth. Does random forest algorithm simply ignore such trees?
Trees in random forests are very deep, and indeed typically grown until the terminal nodes are pure. A lot of these splits are overfit. The overfitting averages out when the predictions are averaged.
You can illustrate this by growing a random forest with only random noise as predictors. The model will be forced to use the noise for splitting.
By the way, a new subset of predictors is randomly selected for each split, not for each tree.