Understanding decision tree concept I am referring Ehthem Alpaydin, 'Introduction To Machine Learning' book.

Under the chapter 'Decision Trees', I need help with understanding the concept of discriminant and how it is being used in this paragraph:


  Each $f_m(x)$ defines a discriminant in the d-dimensional input space dividing it into smaller regions that are further subdivided as we take a path from the root down. $f_m(.)$ is a simple function and when written down as a tree, a complex function is broken down into a series of simple decisions. Different decision tree methods assume different models for $f_m(.)$, and the model class defines the shape of the discriminant and the shape of regions. Each leaf node has an output label, which in the case of classification is the class code and in regression is a numeric value.
In the above image the big square is your $f_m(x)$.

The big square is made up of small partitions. Those partitions are your smaller functions(Partitions) which performs the comparison between your variables and try to find out best possible split.

Here each partition has nearly same labels for classification and nearest numerical values for regression.

The square representation is for 2 dimensions if you want to get for higher dimensions refer the last one with 3 dimensions.

Reference: Elements of Statistical Learning, Basics of Decision trees