Randomness in Random Forests One method to implement randomness in random forests is to use a random subset of the features for a split node.
What will be done at the next split node of the same branch beneath (that current / former split node)? Is it a random subset of the former subset, so, a subsubset? Because I think choosing a subset from the entire feature subset does not make sense?
 A: At each split, you draw a new random sample of $m$ features.
From Hastie et al. Elements of Statistical Learning:

Algorithm 15.1 Random Forest for Regression or Classification.

*

*For $b = 1$ to $B$:

a. Draw a bootstrap sample $Z^*$ of size $N$ from the training data.
b. Grow a random-forest tree $T_b$ to the bootstrapped data, by recursively repeating the following steps for each terminal node of the tree, until the minimum node size $n_\min$ is reached.
i. Select $m$ variables at random from the $p$ variables.
ii. Pick the best variable/split-point among the $m$.
iii. Split the node into two daughter nodes.


*Output the ensemble of trees $\{T_b\}^B_1$.

To make a prediction at a new point $x$:

*

*Regression: $\hat{f}^B_\text{rf}(x) = \frac{1}{B}\sum_{b=1}^B T_b(x).$


*Classification: Let $\hat{C}_b(x)$ be the class prediction of the $b$th random-forest tree. Then $\hat{C}^B_\text{rf} (x) =
> \textit{majority vote} \{\hat{C}_b(x)\}^B_ 1$.


Here's an example. You have a model with $p=4$ features, numbered 0, 1, 2, 3. You've set $m=2$, and at the first split you randomly draw 1,3. Then at the second split, you randomly draw 0,1. Then at the third split you randomly draw 1,3 again. In this example, choosing feature 1 in all three examples is purely a random coincidence. Likewise, so is choosing the same pair 1,3 twice also a random coincidence.
A: Looking at Chapter 15, second edition of Hastie, Tibshirani, Friedman "Elements of Statistical Learning", there is no subsubsetting. All features are available at all steps (by this mean that all features can be in the random feature subset that is used at each step - this is not constrained to those that were in the earlier one). This is well explained in the answer by Sycorax, but I leave this one here for the following:
As there are various schemes around, I can't guarantee that subsubsetting is nowhere done, however I don't see the point of it. I don't see the benefit of constraining randomness in this way. Particularly if some variables are really important and some others are not, it seems to be a waste to constrain later splits to stick to the variables already used before, which ultimately will mean that in the overall ensemble the variables are less uniformly covered, which is not good as we'd like to learn about all of them.
