Gemara

32b (Mishnah): If an Eruv was in a tree below 10 Tefachim, it is valid.

Question: [If he takes it,] he uses the tree!

Answer: The Isur to use a tree is mid'Rabanan. Our Mishnah is Rebbi, who permits Shevus during Bein ha'Shemashos.

33a (Beraisa - Rebbi): If an Eruv was in a tree between three and 10 Tefachim, it is valid, but one may not take it;

If the Eruv was put in a basket and hung on the tree, even above 10 it is valid;

Chachamim say, if the Eruv is in any place from where one may not take it, it is invalid.

Suggestion: Chachamim discuss the Seifa. (It is in a basket hanging on the tree.)

Rejection: If so, they forbid Tzedadim (the sides)!

Answer: They discuss the Reisha.

100a (Beraisa): If roots of a tree are three Tefachim above the ground, one may not sit on them. One may not climb a tree, hang on it or lean on it.

One may not alight (climb a tree) before Shabbos and stay there for all of Shabbos;

(Beraisa #1): If one went up, he may descend [on Shabbos].

Contradiction (Beraisa #2): He may not descend.

Resolution #1: He may descend only if he went up before Shabbos.

Resolution #2: In both cases he went up on Shabbos. He may descend only if he went up b'Shogeg.

Resolution #3: In both cases he went up on Shabbos b'Shogeg. Beraisa #2 fines Shogeg due to Mezid, and Beraisa #1 does not fine.

Resolution #4 (Rav Huna brei d'Rav Yehoshua): Tana'im argue about this:

(Mishnah - R. Eliezer): If blood that requires one Matanah was mixed with blood that requires four, we do four Matanos;

R. Yehoshua says, we do one.

R. Eliezer holds that it is better to do an action. Here also, he would say that it is better to descend. R. Yehoshua holds that it is better to be passive. He would say that it is better to remain on the tree.

Rejection #1: R. Eliezer prefers an action that is a Mitzvah. It is no Mitzvah to descend!

Rejection #2: R. Yehoshua prefers inaction, but not when this is not an Aveirah, e.g. remaining on the tree!

Shabbos 154b: Abaye found Rabah rubbing his son on a donkey's back [on Shabbos].

Abaye: You are using a living animal!

Rabah: I use Tzedadim. Chachamim did not forbid this.

Suggestion: Tana'im argue about Tzedadim:

(Beraisa): [If a tree makes the third wall of a Sukah,] one may not use it on Yom Tov;

R. Shimon ben Elazar permits.

Assumption: The first Tana forbids Tzedadim, and R. Shimon permits.

Rejection (Abaye): No, all forbid Tzedadim. They argue about Tzidei Tzedadim. (Beams are held in holes in the trees, and the Sechach rests on the beams.) The first Tana forbids, and R. Shimon permits.

(Rava): The stringent opinion forbids even Tzidei Tzedadim. The lenient opinion permits even Tzedadim.

The Halachah is, Tzedadim are forbidden, and Tzidei Tzedadim are permitted.

(Rav Ashi): Therefore, one may not rest a ladder against a date tree [and go up]. Rather, he inserts a peg, and rests the ladder against it;

When going up, one may not step on the peg, only on rungs of the ladder.

Rishonim

Rif and Rosh (33b and 10:11): If one went up, one Beraisa permits descending down [on Shabbos], and one forbids. We permit only if he went up before Shabbos. Alternatively, even if he went up on Shabbos, we permit if he was Shogeg.

Rambam (Hilchos Shabbos 21:6): One may not alight on a tree. One may not hang from a tree or lean on it. This is a decree, lest he detach.

Rambam (9): If one went up a tree on Shabbos b'Shogeg, he may descend, but not if he was Mezid.

Magid Mishneh: We permit only the person himself, since he transgresses through staying on the tree. If one put an item on the tree, even from before Shabbos, he may not take it on Shabbos.

Rosh (Shabbos 5:2): R. Shimon ben Elazar permits Tzedadim of a tree on Shabbos. Tzedadim of animals are similar. We learn from here that one may not place a Ner on a tree on Yom Tov (45a). This is not because taking it from there is called using a tree. Rather, we decree lest he alight the tree to take it. Chachamim said only that one may not go up on an animal or tree on Yom Tov.

Poskim

Shulchan Aruch (OC 336:1): One may not alight on a tree. One may not hang from a tree. This is a decree, lest he alight and detach.

Mishnah Berurah (2): Likewise, one may not lean on it.

Kaf ha'Chayim (3): The Gemara forbids also leaning on a tree. This is when it is thin, and it moves when he leans on it. If it is thick, it is permitted.

Mishnah Berurah (3): One may not put something on a tree, take something down from it, or tie an animal to it.

Kaf ha'Chayim (4): All uses are forbidden, lest one detach Peros, leaves or a branch.

Shulchan Aruch (ibid.): If one went up on Shabbos b'Shogeg, he may descend, but not if he was Mezid. If he went up before Shabbos, in any case he may descend.

Beis Yosef (DH Alah): The Rif and Rosh did not bring the answer that the Beraisos argue about whether or not we fine Shogeg due to Mezid. They hold that there is no reason to fine Shogeg due to Mezid. We say that if he went up during the day, even b'Mezid, he may descend. The Magid Mishneh says 'this is if he intended to stay there the entire Shabbos. The Rambam distinguishes Shogeg from Mezid only for one who went up on Shabbos. If he went up before Shabbos, it is permitted in every case.' It seems that this is even if he went up before Shabbos with intent to descend on Shabbos, for when he went up he did not transgress at all. The Tur permits only if he went up with intent to descend before Shabbos.

Mishnah Berurah (7): We fine him, and he may not descend until Motza'ei Shabbos.

Shulchan Aruch (ibid.): Some say that if he went up before Shabbos with intent to remain there after dark, he may not descend then, for he intended to stay there b'Isur.

Mishnah Berurah (10): The Halachah follows the first opinion.

Kaf ha'Chayim (11): Eliyahu Rabah says that the latter opinion is primary. R. Zalman says that the law is mid'Rabanan, so we are lenient. Also, the Mechaber cited the first opinion Stam, and the second opinion is 'some say...'

Rema: If one put an item on a tree from before Shabbos, one may not take it from there on Shabbos.

Magen Avraham (2): For the same reason, one may not place a Ner on a tree on Yom Tov (277:4). The Rosh says that this is not called using something attached, like we permit (317:7) tying a rope on a cow's mouth. Rather, it is forbidden lest he climb the tree to take it. See 628:3. (A Sukah on a tree is Kosher, but one may not go up in it on Yom Tov.) There, it seems that the Rosh forbids taking something from the tree. The Gemara (155a) explicitly forbids taking even from a basket on the tree, and all the more so from the tree itself! Do not say that we forbid the basket lest he alight on the tree. Why should he alight? The basket is below 10 Tefachim! Perhaps he holds that since we decree when it is above 10, also below 10 is forbidden, for we do not distinguish. We do not say so regarding animals, for most of them are not so tall. This is difficult. If so, why did the Rosh say that this is not called using a tree? In any case one may not take an item from there! (Machatzis ha'Shekel - the Rosh is a Posek, and not a Mefaresh. Presumably, he said that it is not due to using a tree, for this affects the Halachah.) According to the Magid Mishneh, we must say that the way to use a tree is to take from it and put on it, but this is not the way to use an animal. Therefore, it is less than Tzedadim. See Rashi (154b DH Tzedadim). We find that one may not tie an animal to a tree (Eruvin 63a), but one may tie a rope on a cow's mouth.

Machatzis ha'Shekel: Rashi says that the one who permits Tzedadim holds that it is abnormal usage. Since we forbid Tzedadim, this is even abnormal usage. However, regarding an animal it is totally unlike the usage, so even we permit.

Mishnah Berurah (12): There is no Isur for a Kli to be on something attached. When he takes it, he uses the tree. Chachamim forbade taking from the tree, for if one does so, he will easily come to lean on it or on a branch. Before Shabbos one may not put on a tree Kelim permitted to use on Shabbos, for he will come to take them down.