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I am aware that categorical variables should be one hot encoded before modeling with random Forests. But I am not entirely sure why.

Lets say we have a predictor categorical variable with 7 levels. The tree should be able to find similarities/differences within this variable if it is numerically encoded. Why do we have to one hot encode categorical variables?

How can I simulate a regression to showcase the difference?

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The problem is the decision tree construction within the random forest. When constructing a DT one selects for all (numeric) attributes all possible (actually seen) values $x$ of that attribute such that making the decision

$$ value < x $$

will generate the most information gain, for all combinations of attributes and values.

If the attribute is categorical, the decision would have to be

$$ value \in subset $$

But that would require an exponential search for all possible subsets of the attribute, which makes the algorithm much less usable.

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