Correlation Beats Strength

Breiman's bound on a random forest's generalization error is

$PE^\ast \le \frac{\bar\rho,(1 - s^2)}{s^2}$

with $s$ the strength of the individual trees and $\bar\rho$ the mean correlation between them.

forest	splits considered at each node	$s$	$\bar\rho$
A	all features (bagging only)	0.55	0.40
B	a random $\sqrt{p}$ subset	0.50	0.20

Which forest has the lower bound?

Neither can be scored without the number of trees, since averaging more trees drives the bound down.

B, at 0.60 against A's 0.92: halving the correlation between the trees more than pays for the drop in individual strength.

A, because its trees are individually stronger and the bound is dominated by $s$ .

They tie: B gives up exactly as much strength as it gains in decorrelation, which is what makes the tradeoff a wash.