Step 1: Understanding the Concept:
The number of branches (Shakhas) of each Veda is traditionally recorded in texts like the 'Mahabhashya' of Patanjali and the 'Charanavyuha'.
The expansion of Vedic literature led to many regional and familial schools.
Step 2: Detailed Explanation:
Let us analyze the statements:
Statement A: Samaveda has one thousand branches.
Patanjali in the 'Paspshahnika' states: "sahasravartmā sāmavedaḥ".
(The Samaveda has a thousand paths/branches).
Thus, Statement A is correct.
Statement B: Yajurveda has one hundred and one branches.
The traditional count for Yajurveda is: "ekaśatam adhvaryuśākhāḥ".
(There are 101 branches of the Adhvaryu/Yajurveda).
Thus, Statement B is correct.
Statement C: Atharvaveda has fifteen branches.
According to the 'Charanavyuha' and Patanjali, the Atharvaveda has nine branches ("navadhā ātharvaṇo vedaḥ").
Fifteen is the traditional number for the branches of the Shukla Yajurveda (Madhyandina, Kanva, etc.).
Thus, Statement C is incorrect.
Statement D: Samaveda has thirty branches.
As established in Statement A, the traditional number is 1000.
Thirty is not a standard traditional count for Samaveda.
Thus, Statement D is incorrect.
Conclusion: Statements A and B are correct.
Step 3: Final Answer:
Statements A and B are correct. Thus, the correct option is (2).