Why Random Forests Work, Measured: The Correlation Identity and Breiman's Strength-Correlation Bound on Wine Quality and Gamma-Ray Data

A random forest averages many decorrelated trees. Two results explain why this works: the variance of an average of B trees with pairwise correlation rho is rho times the tree variance plus (1 minus rho) over B times the tree variance, and Breiman's bound limits the forest's error by the mean correlation of the trees divided by the square of their strength. Both are usually quoted, rarely measured. I prove both and measure every quantity on two public datasets. On UCI Wine Quality (white, 4898 wines), I train 120 forests of 200 trees on independent training sets of 1000 wines and evaluate on 300 fixed test wines. The correlation between trees across training sets is 0.043 to 0.078, rising with the number of features tried at each split, and the identity predicts the forest's variance within 11%. Averaging removes 91% to 96% of a single tree's variance and halves its test error, from about 1.07 to 1.14 down to 0.53 to 0.55. On MAGIC gamma telescope data (19,020 events) the mean correlation of tree margins rises from 0.26 with one feature per split to 0.37 with all ten, while strength rises from 0.54 to 0.61. The Breiman bound sits between 0.60 and 0.64 against actual errors of 0.124 to 0.130: it orders the settings correctly at the extremes but is five times too loose to guide tuning. The lowest test error, 0.1236, comes from trying two features per split.