Does it make sense to have 2 branches resulting in same decision while using one single variable to divide the predictor multiple times? I have been recently come across a problem that entails using a decision tree that only uses one continuous variable to divide the predictor on multiple threshold, while some splits result in the same decision. The decision tree is as follows:

So I have two questions:

*

*Does it make sense to use one variable multiple times for dividing the predictor?

*Is it possible that two splits result in the same decision?

I didn't share any codes or data set because I just want to know whether this condition is possible and why we would prefer this.


Does it make sense to use one variable multiple times for dividing the
predictor?

Yes, if the variable is useful and has different regions that are not connnected, it can be used mutliple times, e.g. return 1 if 1<x<4 else 0 would require two splits.

Is it possible that two splits result in the same decision?

It is possible as well. Although the final decision is the same, which might have been obtained by taking the majority of cases in each branch, the overall entropy might improve with the branching.
For example, if the distribution is 5-7 for class 0 and 1 respectively, and after the split it becomes (2-3) and (3-4) in the branches, the information gain would be:
h  = -(log(5/12) * (5/12) + log(7/12) * (7/12)) 
b1 = -(log(2/5) * (2/5) + log(3/5) * 3/5)
b2 = -(log(3/7) * (3/7) + log(4/7) * 4/7)
h - (b1 * 5/12 + b2 * 7/12)

which would result in the following gain
0.000408677

