Haircut for a Child During the Thirty Days of Mourning for His Father

Is it permissible to cut the hair of a young child during the thirty days of mourning for his father?

It is permitted. See in the source the opinion of those who prohibit it.

In the Gemara Moed Katan, page 14: "Shmuel said: A small child born during the festival is permitted to have a haircut during the festival, as there is no greater imprisonment than this. During the festival - yes, initially - no. Rabbi Pinchas objected: All those who are permitted to have a haircut during the festival are permitted to have a haircut during the days of mourning. Those who are forbidden to have a haircut during the festival are forbidden to have a haircut during the days of mourning. And if you say there is a dispute regarding the child, it turns out that mourning applies to a child. But it was taught: they tear the garment for a child due to emotional distress! - Rav Ashi said: Does it state that this is forbidden? Perhaps some are forbidden, and some are permitted. Ameimar, and if you wish, Rav Shisha, son of Rav Idi, teaches thus: Shmuel said: A small child is permitted to have a haircut during the festival, whether born during the festival or initially. Rabbi Pinchas said: We also taught: All those who are permitted to have a haircut during the festival are permitted to have a haircut during the days of mourning. Those who are forbidden to have a haircut during the festival are forbidden to have a haircut during the days of mourning. If you say a child is forbidden, it turns out that mourning applies to a child. But it was taught: they tear the garment for a child due to emotional distress. - Rav Ashi said: Does it state that this is forbidden? Perhaps some are forbidden, and some are permitted." It is clear that there is no prohibition to cut the child's hair during mourning, and they do not tear for him except due to emotional distress.

This is also ruled by the Beit Yosef in Siman 389 in the name of the Laws of Joy, and similarly in the Mordechai in the Laws of Mourning, end of Siman 499.

The rabbis are divided on whether there is education in mourning for a child, the opinion of the Ri"tz Gaiot and the Taz and the Bach, that there is a law of education in mourning for a child. And they hold that the law explained in the Gemara that they tear due to emotional distress is for a child who has not reached the age of education, but for a child who has reached the age of education, they tear by law.

However, in the Shach, Siman 384, Subparagraph 5, it is written that there is no law of mourning for a child, and so it is written in the Beit Hillel in Siman 389.

In practice, they adhere to the main law, that there is no education for a child in mourning.