The standard DECISION-TREE-LEARNING algorithm described in the chapter does not handle cases in which some examples have missing attribute values.
1. First, we need to find a way to classify such examples, given a decision tree that includes tests on the attributes for which values can be missing. Suppose that an example $\textbf{x}$ has a missing value for attribute $A$ and that the decision tree tests for $A$ at a node that $\textbf{x}$ reaches. One way to handle this case is to pretend that the example has all possible values for the attribute, but to weight each value according to its frequency among all of the examples that reach that node in the decision tree. The classification algorithm should follow all branches at any node for which a value is missing and should multiply the weights along each path. Write a modified classification algorithm for decision trees that has this behavior.
2. Now modify the information-gain calculation so that in any given collection of examples $C$ at a given node in the tree during the construction process, the examples with missing values for any of the remaining attributes are given “as-if” values according to the frequencies of those values in the set $C$.