The algorithm here is rather straightforward. Assume that our weak classifier is a decision tree and we consider a binary set of outputs with y_i \in \{-1,1\} and i=0,1,2,\dots,n-1 as our set of observations. Our design matrix is given in terms of the feature/predictor vectors \boldsymbol{X}=[\boldsymbol{x}_0\boldsymbol{x}_1\dots\boldsymbol{x}_{p-1}] . Finally, we define also a classifier determined by our data via a function G(x) . This function tells us how well we are able to classify our outputs/targets \boldsymbol{y} .
We have already defined the misclassification error \mathrm{err} as
\mathrm{err}=\frac{1}{n}\sum_{i=0}^{n-1}I(y_i\ne G(x_i)),where the function I() is one if we misclassify and zero if we classify correctly.