Cost Complexity Pruning

We consider a sequence of trees alongside some tuning parameter $\alpha$ and wish to minimize the following

$$\sum _{m=1}^{|T|}\sum _{i:x_i\in R_m}(y_i-{\hat{y}}_{R_m}{)}^2+\alpha |T|$$

$|T|$ is the number of terminal nodes $R_m$ is the terminal node region

If we set $\alpha =0$ then we will have $T_0$. $\alpha$ scales with the number of terminal nodes, $|T|$ so there is a price to pay for having many terminal nodes.

We can then obtain a sequence of subtrees as a function of $\alpha$. (For some $\alpha$ we would get the best subtree)