GEMARA

9b - Mishnah: A Sukah under a tree is (Pasul) like a Sukah in the house.

Rava: This is if the tree gives more shade than it lets sun through. If it allows more sun than shade, it is Kosher.

Inference: I learn from the Mishnah. Why does it say 'it is like a Sukah in the house?' Let it say 'it is Pasul'!

Answer: The Mishnah compares it to a house to teach that it discusses a tree similar to a house, i.e. it gives more shade than sun.

Question: Why is it Kosher when it allows more sun than shade? Pasul Sechach joins with Kosher Sechach!

Answer (Rav Papa): The case is, Chavtan. (Meforshim argue about whether this means that he separated the Pasul (attached) Sechach from the Kosher Sechach, or cut what was attached, or lowered the branches and mixed them with the Kosher Sechach, or knocked the leaves off the branches.)

Question: If so, obviously it is Kosher!

Answer: One might have thought that we decree to forbid, lest he not do Chavatah.

Question: We already learn this from a Mishnah!

11a - Mishnah: If one draped vines over a Sukah, it is Pasul. If there was more Sechach than vines or if he cut them, it is Kosher.

Question: If he did not do Chavatah, why it is Kosher? Pasul Sechach joins with Kosher Sechach!

Answer: We must say that he did Chavatah.

This shows that we do not decree lest he not do Chavatah

Answer: One might have thought that b'Di'eved it is Kosher, but l'Chatchilah we decree to forbid. Rava teaches that this is not so

Beraisa: "Ba'Sukos" is written without a Vov, to teach that one must be under only the Sukah. The Sukah may not be under another Sukah or a tree, or in the house.

RISHONIM

Rif: The Gemara established our Mishnah to be when he did Chavatah. If not, Pasul Sechach joins with Kosher Sechach, and it is Pasul.

Rambam (Hilchos Sukah 5:12): A Sukah under a tree is (Pasul) like a Sukah in the house. If one draped leaves or branches of trees over a Sukah and put Sechach on them and later cut them:

If there was more Sechach than them (what was on the tree), it is Kosher;

If not, one must shake them after cutting them in order that they will be l'Shem Sukah.

Ra'avad (13) and Magid Mishnah: The Rambam explains that 'Chavtan' means 'he separated (the Pasul Sechach from the Kosher Sechach)'.

Magid Mishnah: He teaches that if there is more Kosher Sechach there is no need to shake the vines after cutting them. This is obvious! Since there was more Kosher Sechach, it was enough to separate it, and cutting is even better. The need to shake it is only for "Ta'aseh" v'Lo Min ha'Asuy.

Lechem Mishneh: Even though 'under a tree' connotes that the Sukah is distanced from the tree, the Gemara assumes that they are together, for otherwise there is no Chidush, like it asks later.

Question: Why doesn't the Rambam distinguish whether or not the tree has more sun or shade?

Answer (Lechem Mishneh): The Rambam holds that really, it depends only on whether most of the Sechach is Pasul or Kosher.

Kesef Mishneh: Pirush ha'Mishnayos clearly explains that 'Chavtan' means 'cut'. It appears that also here the Rambam explains this way. The Mishnah (11a) is Machshir if there is more Kosher Sechach; the Gemara (9b) says that this is only if Chavtan, i.e. the vine was (merely) cut. Alternatively, (if there is more Pasul Sechach) it suffices to (properly) cut the vine, i.e. to cut and shake it. Regarding a Sukah under a tree, if the tree is mostly sun presumably there is more Kosher Sechach, therefore Chavtan suffices. If the tree is mostly shade presumably there is more Pasul Sechach, therefore the branches must be cut and shaken. Even though the branches fall after being cut, they must be directly shaken.

R. Mano'ach: Chavtan means to beat the branches until most of the leaves fall off. Ri'ag forbids to sit under the Pasul Sechach, even when the majority and the Sukah are Kosher. In this case Tosfos permits to sit under the Pasul Sechach, but one may not sit under air.

Ran (5a DH v'Nimtza) and Magid Mishnah citing Ramban: There are three cases:

If Kosher Sechach is under Pasul Sechach it is always Pasul. However, if one would remove the Kosher Sechach under the Pasul and it would still give mostly shade it is Kosher, as long as there are not four Tefachim (of Pasul Sechach) in one place.

If one knocked down the Pasul Sechach and combined them, if the Kosher Sechach was mostly shade and the Pasul Sechach was mostly sun, the Pasul is Batel in the majority and it is Kosher. If not, it is Pasul.

If one used Kosher Sechach next to Pasul Sechach, if half is Kosher the Sukah is Kosher (like the Mishnah 15a).

Rosh (1:14): Rashi does not require mixing when the Kosher Sechach gives a majority of shade. Then we simply ignore the Pasul Sechach. However, Avi ha'Ezri requires Chavtan in any case, for otherwise one is in the shade of the tree. This is whether the tree or Sukah was there first. However, if the branches are over the air between the Kosher Sechach it is Kosher.

Hagahos Maimoniyos (70): Rashi (10a DH Mahu d'Seima Nigzor) retracted and agrees with Avi ha'Ezri.

POSKIM

Rema (OC 626:1): One should not make a Sukah under a tree.

Shulchan Aruch: If one did, some say that if the tree gives mostly shade it is Pasul in any case.

Kaf ha'Chayim (7): Even Chavatah does not help, even if the Sechach gives more shade than the tree, even if the tree is very high above the Sechach or less than 10 Tefachim above it, it is considered two Sechachim.

Shulchan Aruch (ibid): If the tree allows more sun than shade, and the Sukah itself gives mostly shade without the tree, it is Kosher, even if the branches were not lowered and mixed with the Sechach. If the Sukah gives mostly shade only with the tree, one must lower the branches and mix them with the Sechach so they are not recognizable. We need a majority of Sechach, and then the branches are Batel to the Sechach.

Question: Normally, l'Chatchilah we may not Mevatel an Isur!

Answer #1 (Levush, cited in Taz 2): Normally we do not Mevatel an Isur lest one permit the entire Isur. Here he intends to do a Mitzvah, we are not concerned lest he do a Mitzvah through an Aveirah.

Rebuttal (and Answer #2 - Taz 2): Decrees were made, e.g. not to use boards for Sechach. The question does not begin, for there is no Isur here. Even during Sukos it is not Asur to sit in a Pasul Sukah. Rather, one who does so does not fulfill the Mitzvah.

Answer #3 (Taz 2 citing Mordechai): Ein Mevatlin Isur is mid'Rabanan, and it applies only to benefit. Mitzvos are not for benefit. Similarly, one may Mevatel Chametz before Pesach, for it is not Asur.

Answer #4 (Magen Avraham 3): Perhaps indeed this is permitted before Sukos, but on Chol ha'Mo'ed it is forbidden because of Mevatel Isur!

Answer #5 (Magen Avraham 3, according to Machatzis ha'Shekel DH mi'Kol): Since there is enough Sechach by itself, we do not need the branches, so there is no intent for Bitul. However, it is not clear how to answer for the opinion that allows the branches to joins with the Sechach to create a majority of shade.

Some say that even if the tree allows more sun and the Sukah itself gives mostly shade without the tree, if the branches are directly over the Sechach it is Pasul, whether the tree was there first or the Sukah was there first.

Gra (DH Bein): The Ba'al ha'Ma'or (5a) disqualifies only if the tree was there first. This is difficult, for the Gemara did not specify. Also, it should have answered that the Chidush is that we do not decree lest one use such a Sukah when the tree was there first!

Rema: However, if one lowered the branches and mixed them with the Sechach so they are not recognizable, they are Batel and it is Kosher. Similarly, if one put Kosher Sechach on the Pasul Sechach this is a mixture, and it is Kosher.

Magen Avraham (4): Since we discuss when the Sukah itself gives mostly shade, we do not need Bitul, so we are not concerned whether or not the branches are recognizable.

Shulchan Aruch: If the branches are over the air between the Kosher Sechach (Rema - or if there is so much Kosher Sechach that even if the Sechach under the tree was removed a Shi'ur would remain) it is Kosher.