Gradient Boosted Decision Trees How to Find Prediction of Each Tree?
Sklearn's GradientBoostingClassifier is not implemented using trees of DecisionTreeClassifiers. It uses regressors for both classification and regression. You can read it here: 


  GB builds an additive model in a forward stage-wise fashion; it allows
  for the optimization of arbitrary differentiable loss functions. In
  each stage n_classes_ regression trees are fit on the negative
  gradient of the binomial or multinomial deviance loss function. Binary
  classification is a special case where only a single regression tree
  is induced.


This means it will not be as simple as calling predict on the tree estimators. I previously suggested it could be implemented using sklearn's private methods, but as BenReiniger pointed out sklearn has already implemented this for us in the method staged_predict.