Q:

Relation between calculated probabilites and Decision Tree

I'm currently building a classification model in MS azure with the Two-Class Boosted Decision Tree algorithm.  From my basic knowledge I know that the decision tree splits the features by a cut value in two or more leafs.  Thus, when classifying new set of data, you just run through the tree according to the splits. 
What I'm missing at this point is, how the probability of a certain outcome can be calculated in this classification process.  When scoring the model with the test set, azure calculates the probability of each row of having a positive (=1) outcome.  
Can someone explain me the association here between the decision tree with having rules assigning a clear outcome for every row but on the other hand probabilites are calculated for every row, which would not be not a clear outcome depending on the general cut-off value?  
Thanks a lot! 
This is part of the column in the scoring where probabilities for each row in the test set is calculated. 

A:

Most classification decision tree algorithms provide both a class and probability.  The probability that you see is the number of correct classifications at each leaf.  So if at one leaf in your tree you have 1 observation of class A and 3 of class B you will see a probability of .75 for class B and the model will predict class B. You can read this slideshow for more details on decision trees http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15381-s06/www/DTs.pdf 

