In the following lectures Tree Methods,
they describe a tree algorithm for cost complexity pruning on page 21.
It says we apply cost complexity pruning to the large tree in order to obtain a sequence of best subtrees, as a function of $alpha$. My initial thought was that we have a set of $alpha$ (i.e. $alpha in [0.1, 0.2, 0.3])$. And then we compute the K-fold cross-validation for each set $alpha$ and choose the $alpha$ corresponding to the lowest K-fold cross validation.
For reference, the K-fold cross-validation that I am referring to is described on slide 15 in the following Cross-Validation.
However, after some thinking and reading; I found there is a theorem that the sequence of subtrees went through by weakest link pruning will contain the subtree $T_alpha$.
Where $T_alpha$ is the subtree that minimizes the cost complexity criterion
$$
C_alpha (T) = sum_{m=1}^{|T|} sum_{i: x_i in R_m} (y_i – hat y_{R_m})^2 + alpha|T|
$$
described on slide 19 in the Tree Methods slides.
Since this theorem exists I thought it might be plausible that you can deduce a neat mapping of each $alpha in mathbb{R}$ to the optimal subtree $T_alpha$. Or at least we can see that for a certain interval of $alpha$s it will correspond to a specific subtree. I found this theorem on page 308 of the Elements of Statistical Learning book of print 4. It would be great if someone could clarify the algorithm and it would help to know if such a mapping exists.
