"V'hisho'anu tachas ho'EITZ" - And repose under the tree
Bereshis 18, 4 Avrohom spoke to the three men (angels)
Rashi says "tachas ho'ilon." Sifsei Chachomim explains that Rashi is forewarning that EITZ is actually wood, and thus Eitz seems to indicate a tree that was felled. ILON is a live tree. If so, why doesn't the verse say ILON? If we posit that the word ILON is not a Loshon Hakodesh word for a tree, and only EITZ is (as seems to be indicated by the fact that we do not find ILON in Tanach except in one place, Daniel 4:7, where the language used is Aramaic), it is understood why EITZ was used in the verse. Thus, the word "ho'eiloh" in Breishis 35:4 does not mean a tree, but rather is the name of a specific type of tree. Those who say that the word "ho'allone" in verse 8 is a tree, again it is the name of a specific type of tree (see Targum Yerushalmi Hasholeim in Chumash Otzar Rishonim). We then would need to understand what Rashi is teaching us, as EITZ also means a live tree.
Alternatively, Hadar Z'keinim says on the words "v'hu omeid a'lei'hem tachas ho'eitz" (verse 8), that this means that Avrohom stood above them IN PLACE OF a tree, i.e. Avrohom used his body to block the sun and cast a shadow upon his three guests. If we say that Avrohom had this in mind in our verse, he used the word EITZ to connote a (non-growing from the ground) tree to indicate that he would be the tree.
The Zohar states that this tree is the Tree of Knowledge mentioned regarding Gan Eden.
The Kli Yakar asks why does the Torah say “under the tree” and not “by the tree”. He suggests that Avrohom was alluding to the concept that the guests should understand that we all exist in the shade of Divine Providence, the Provider of all our needs.
The Medrash Rabbah (48,10) says that in the merit of Avrohom's saying "v'hisho'anu tachas ho'eitz" the bnei Yisroel would later merit to have the mitzvah of "basukkos teishvu shivas yomim" “in sukkos you shall dwell for seven days” (Vayikrah 23:42). To allude to this concept the verse might have specifically used the word EITZ to indicate the mitzvah of sitting in a sukkah under the covering of wood, and not a live tree, as this would invalidate the sukkah. Rabbi Shmuel of Sanot goes so far as to say that Avrohom actually built his guests a sukkah to protect them from the direct heat of the sun.
Nachalas Tzvi explains the connection between the action of Avrohom and the reward of the mitzvah of Sukkah. The main concept of Sukkos is for a person to have faith in HaShem and dwell within His shade. We need to realize that only under His Providential care does one exist. And this is the message involved in Aggadah (Avodah Zarah 3a) that HaShem offered the mitzvah of Sukkah to the non-Jews at the time of Moshiach.
Further, we exist in the shade of HaShem, and not in the heavenly structures. Thus, Avrohom was alluding for his guests to forsake avodah zarah (worship of the heavenly objects) and worship the true and only G-d.
Thus, HaShem rewarded Avrohom that his descendants would have a mitzvah which would continue to teach this concept of existing under the shade of Divine Providence, and not due to the objects in the Heavens. This is according to Tehillim 145, 19, רצון יראיו יעשה.
