How to Validate Decision Tree model by using *statistical tests*?
I am assuming you are using the D.T. for binary classification for a moment. One of the first tests I learned (and still dam good) is the 2x2 contingency table or frequency table or marginal frequencies or 2-way tables (many names means it's been around a while). It is simple to use and can branch off into many other tests and areas. Such as;  


Phi Coefficient of Association
Chi-Square Test of Association
Fisher Exact Probability Test
Accuracy and precision
F-score or F-measure, ...(to name a few)


I usually like Kahn Academy way over Wikipedia(hate their explanations).