Would decision tree propagate error? Given a regression dataset $X,Y$. Suppose there are two different decision tree (CART) $T_1, T_2$ fitted from it. Each using different feature encoding method. And we get two different tree.
And there are many ways to compare them. The simplest might be have k fold cross validation.
For each tree, if we construct a “mini” tree by only using the root decision node and construct a decision stump out of it: $S_1, S_2$.
Now my question is for whatever reasonable comparing method we choose, would the outcome of $T_1$ and $T_2$ be consistent to if we apply to $S_1$ and $S_2$ ?

I don't think so. Based on the feature encoding used, the root split might be better for $T_1$ but not enough for $T_2$, because the consequent levels of the tree may complement the root very well. For example, if $T_1$ uses a 1D mapping for an inherently multi dimensional feature, the first split might be very informative, especially compared to splitting a the one-hot encoded version of the same feature. So, $T_1$ may look like the better option, but if the feature is best represented on a multi dimensional setting, the following splits in the tree in $T_2$ may compensate it and can be better than the one in $T_1$.
