Understanding decision tree concept
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