Gemara

32b - Mishnah: If the top of a Hadas was Katum (cut off), it is Pasul.

33a - Ula bar Chinena - Beraisa: If the top of a Hadas was cut and a date grew on top, it is Kosher.

Question (R. Yirmeyah): If the top of a Hadas was cut before Yom Tov and a date grew on top on Yom Tov, what is the law?

This question is not resolved.

33b - Mishnah: If the top of an Aravah was Katum, it is Pasul.

34b - Mishnah - R. Yishmael: We take two Aravos, one Lulav, one Esrog and three Hadasim. Two of the three may be Ketumim;

R. Tarfon says, all three may be Ketumim.

R. Akiva says, we take one of each species.

Question: If R. Yishmael requires whole Hadasim, he should require three whole ones. If Ketumim suffice, he should allow three Ketumim!

R. Ami: R. Yishmael retracted.

Shmuel: The Halachah follows R. Tarfon.

This is consistent with what Shmuel said elsewhere. He threatened Hadasim merchants that if they charge too much, he will expound that the Halachah follows R. Tarfon.

Rishonim

The Rif brings the cited Mishnayos.

Rif (15b): The Halachah follows R. Tarfon, therefore a Niktam Hadas is Kosher even if a date did not grow.

Rebuttal #1 (Ra'avad, Hilchos Lulav (cited in Migdal Oz Hilchos Lulav 8:5) DH Inyan Niktam Rosho): A Mishnah of clearcut Halachah disqualifies Niktam Rosho. The entire Sugya connotes that it is Pasul. The Mishnah of 34b agrees. It does not discuss Niktam Rosho, rather, Ketumim. These are upper branches cut off (from the bottom) while they are soft. The Yerushalmi suggested that Ketumim are preferable; later, it retracted. R. Yishmael requires one branch that was not cut at all, and allows two Ketumim. It is far easier to find three Ketumim that are Avos than one uncut branch that is Avos (covered by three leaves at every juncture).

Rebuttal #2 (Ba'al ha'Ma'or, according to Ran 16b DH Masnisin): Ketumim are Hadasim from which were cut off branches going out to the side. A typical Hadas has branches going out to the side. There are not three together, so they interrupt the 'Avos' of the Hadas, so they must be cut to make the Hadas Avos. This is why it is hard to find a Kosher Hadas that is not Katum. R. Yishmael and R. Tarfon argue about such Ketumim.

Defense (Milchamos Hash-m, and Ramban's Hasagos on Hilchos Lulav of the Ra'avad): All the Ge'onim ruled like the Rif. Hadasim are "Anaf Etz Avos"; this connotes branches cut off the tree! It is not more Hadar or more considered Hadas when it is uncut. The top leaves dry up and fall off, therefore it is far easier to find a branch that is Avos if it is Niktam Rosho than if the top is intact.

Rebuttal (Ran, ibid.): It is unreasonable to say that the Sugya of Dichuy (33a, when a Hadas was Niktam and a date grew) is not according to Halachah! The Ramban says that only R. Yirmeyah (there) disqualifies Niktam Rosho. The other Amora'im answered him according to his reasoning. This is a weak answer.

Rambam (Hilchos Lulav 8:5): If the top of a Hadas was cut it is Kosher.

Rebuttal (Ra'avad): It is Pasul. Hash-m withheld the proper understanding of this from earlier Chachamim and gave me Ru'ach ha'Kodesh to understand it (see Ra'avad above).

Magid Mishnah: The Ramban refuted the Ra'avad's proofs. Nevertheless, it is good to be concerned for the Ra'avad's opinion.

Rambam (Hilchos Lulav 8:6): If the top of an Aravah was cut it is Kosher.

Ra'avad: Aravah is like Hadas.

Magid Mishnah: Even though the Mishnah (33b) disqualified Katum, the Rambam holds that since the Halachah follows R. Tarfon, that Katum does not disqualify Hadas, the same applies to Aravah. The Rif and Ritz Gei'us disagree. The Ra'avad holds that Katum disqualifies Hadas, and also Aravah.

Rosh (3:14): The Halachah follows R. Tarfon. The Ra'avad and Ba'al ha'Ma'or distinguish between Niktam and Katum; this is very difficult.

Question: Why is Niktam Rosho Kosher for Hadas, but not for the other Minim?

Answer #1 (Rosh citing R. Yeshayah and Magid Mishnah 8:6 citing Ramban): The beauty of Hadas is that its leaves cover the wood; it is not evident when the top is Katum. Katum is evident in the other Minim, so they are not Hadar

Answer #2 (R. Mano'ach Lulav 8:5): Hadar of Hadas is being Avos (covered with leaves), therefore we are not concerned for a cut top. Hadar of the other Minim is in the branch itself.

Answer #3 (R. Shmuel, brought in R. Mano'ach, ibid.): Hadar can sprout a date after Niktam Rosho, therefore it is as if it was not cut.

Hagahos Ashri: The Halachah follows R. Akiva; even if there was one Hadas and it was not Katum, it is Kosher. Avi ha'Ezri agrees, for the Gemara asked why Shmuel did not threaten to expound like R. Akiva.

Tosfos (34b DH v'Lidrosh): The Gemara inferred from Shmuel's threat that he holds like R. Tarfon. If he would threaten to expound like an opinion that is not the Halachah, he would have threatened to expound like R. Akiva, who is even more lenient! The Gemara rejects this, for R. Akiva is more stringent (most Hadasim are Ketumim). Until now, Shmuel did not expound like R. Tarfon because it is preferable not to use Ketumim.

Poskim

Shulchan Aruch (OC 646:1,10): If the top of a Hadas was cut it is Kosher, even if a date did not grow there. Some say that it is Pasul.

Mishnah Berurah (30): If a date grow there it is surely Kosher because the cut is not visible.

Kaf ha'Chayim (56): It is Kosher because the cut is not visible. If there are no leaves covering the cut top, one must cut more until the top is covered by leaves. If not, it is Pasul.

Rema: It is good to be stringent when possible. It is called Niktam only if the wood was cut.

Taz (1): The Re'em says that R. Tarfon allows Ketumim b'Di'eved. The Stam Mishnah (32b) says that it is Pasul l'Chatchilah (one should try to get one that is not Katum). The Mordechai says that the Mishnayos do not distinguish Niktam Rosho between Lulav and Hadas. Therefore, in both cases l'Chatchilah one should get one that is not Niktam, but if one cannot it is Kosher.

Mishnah Berurah (31): Perhaps the stringent opinion is stringent only if all three are Ketumim. The Shulchan Aruch (651:1) allows two to be Ketumim in pressed circumstances.

Kaf ha'Chayim (59): In pressed circumstances one may rely on the Rif, Rambam and Rosh who allow even three Ketumim.

Mishnah Berurah (33): If small branches exude between the junctures of leaves one must cut them. All agree that the Hadas is Kosher.

Shulchan Aruch (OC 647:2): If the top of an Aravah was cut it is Pasul. The Rambam says that it is Kosher.

Question: How can the Rambam rely on reasoning to overturn a Stam Mishnah, against which no one argued? Also, if so he should allow a Katum Lulav!

Answer #1 (Taz 4): He learns from the order of the Mishnayos. First the Mishnah (29b) discusses Lulav, then Hadas (32b), then Aravah (33b), then the number of each Min and the argument about Hadasim Ketumim (34b), then Esrog (34b). Seemingly, the number of each Min and the argument about Ketumim should have been taught right after Hadasim, or after all four Minim! Rather, it was put after Aravah to teach that R. Tarfon permits also Aravos Ketumim.

Answer #2 (Gra DH veha'Rambam): In Aravah and Hadas, Niktam Rosho refers to the wood. (In a Lulav it applies to the leaves.)

Kaf ha'Chayim (17,21): We hold that it is Pasul. Therefore, if the Aravah is too long one must trim it from the bottom.