Is the decision tree greedy algorithm correct?

Problem definition:
According to the greedy idea of ​​large information gain, can the constructed tree be guaranteed to be the smallest?

Answer:
No

for example:

Item: abcdefgh
Category: 1 1 1 1 0 0 0 0
Feature 1: 1 1 0 0 1 0 0 0
Feature 2: 1 1 0 0 1 1 0 0
Feature 3: 0 0 1 1 0 0 1 1

According to greed, feature 1 should be selected for splitting, which has the largest information gain, but this way the subtree needs three layers.
But feature 2 + feature 3 is the smallest tree, with only two levels.