If we consider a full grown decision tree (i.e. an unpruned decision tree) it has high variance and low bias.
Bagging and Random Forests use these high variance models and aggregate them in order to reduce variance and thus enhance prediction accuracy. Both Bagging and Random Forests use Bootstrap sampling, and as described in "Elements of Statistical Learning", this increases bias in the single tree.
Furthermore, as the Random Forest method limits the allowed variables to split on in each node, the bias for a single random forest tree is increased even more.
Thus, the prediction accuracy is only increased, if the increase in bias of the single trees in Bagging and Random Forests is not "overshining" the variance reduction.
This leads me to the two following questions: 1) I know that with bootstrap sampling, we will (almost always) have some of the same observations in the bootstrap sample. But why does this lead to an increase in bias of the individual trees in Bagging / Random Forests? 2) Furthermore, why does the limit on available variables to split on in each split lead to higher bias in the individual trees in Random Forests?