How does Random Forest relate to the bias-variance-covariance decomposition formally?

Random Forest reduces ensemble variance by lowering covariance between trees while averaging individual variances.

The ensemble error can be decomposed as Var(average of trees) = (1/T)σ² + (1 - 1/T)ρσ², where σ² is variance of individual trees and ρ is pairwise correlation. Random Forest explicitly targets ρ via bootstrap sampling and feature subsampling. Even if σ² remains high (deep trees), reducing ρ yields significant variance reduction. This decomposition explains why RF works even with overfitted base learners.