Avraham told his guests והשענו תחת העץ to recline in the shade under the tree. This suggestion of Avraham is not only a message to them as part of his offering them the physical comforts of Hachnasas Orchim. It also contains an instruction of how to align one's life with the will of Hashem making everyday living more meaningful and fulfilling.
The Navi says (Yeshaya 65,22) כימי העץ ימי עמי the days of a tree parallels the days of My People. Am Yisrael will have longevity more than any other nation on earth. Let us suggest another interpretation to these four words. Chazal tell us that when the neshama of a departed comes to the Bais Din in Shamayim, they will inquire from him if he was קבעת עתים לתורה Did he fix and set aside time for learning the Torah. My father ע"ה told me many times a different explanation of this heavenly inquiry. In the Heavenly Court they will ask Did you during the span of your lifetime set the "times" to the laws and spirit of the Torah or did you allow the Torah Life style to become outdated and obsolete. You rather chose to make adjustments and compromises in your observance to the Torah in order for you to remain in your comfort zone together with the seductive cultural times and demanding innovations that you must acquire and experience.
This is what the Navi is saying, כימי העץ is referring to the Torah which is called .עץ החיים The Torah is very makpid and strict to ensure one is in synch with exact timing. There is zman Neitz, Kerias Shmah, Tefillos, Shabbos, Rosh Chodesh, Yom tov, how long one must wait after tefillah before he steps forwards, how long does one have to sleep which would require him to wash his hands, how much waiting after finishing a meal is one still allowed to bentch, Hilchos Shemitta, Yovel, Sefirah, etc.
So too ימי עמי Am Yisrael's days are negated to the fulfillment of mitzvos within time zones and limitations in everyday life. And in the broader sense, even the fashionable and new trends of תשפ"ה times are only acceptable if they align completely with the Torah. The galus only came upon us when our days were not in sync with what the Torah expects from our days, and instead we adapted and altered the Torah to fit the new vogues and fads. This is gleaned from the gematriah of כימי העץ ימי עמי 425 which adds up to משיח בן דוד עה"כ 425. Had this equation between Am Yisrael and the Torah been fulfilled which is gematriah כדת 425, Moshiach would have already been here a while ago.
The Torah begins with the first word of בראשית . Rashi writes that this word alludes to Torah since Torah is called .ראשית It was created before anything else in the world since the Torah served as the blueprints of the universe. Another interpretation of the word בראשית is that Hashem created the beginning of time since time necessitates a beginning. The Vilna Gaon explains that the Baruch Shearmar tefilla which is the commencement of Pesukai D'zimra, praises Hashen for ברוך עושה בראשית which is referring to the creation of time. Therefore, the product outcome of the word בראשית when synthesizing these two interpretations, is that it was the Torah that created time. Being so, one must ensure in every generation no matter how technological progressive, to subjugate the times to the Torah its creator, and not vice versa.
This is the lesson that Avraham was teaching his guests when he sat them down under the tree והשענו תחת העץ. The word for clock and watch is שעון whose root word is שעה time. Time can be used correctly and elevated by filling it with acts of sanctity, holiness, purpose, and meaning. Or one can use the three letters of שעו for acts ofעשו , emptiness, decadence, wastefulness, unconstructiveness etc. Avraham was telling his guests that there is a very elusive element in creation called time. It is not tangible and therefore people are very unaware of its preciousness. However the reality is that an unproductive spiritual day is a loss of a colossal fortune of eternity which cannot be returned.
By Avraham and Sarah the Torah states that they were זקנים באים בימים that even at old age they were coming older with all their days not losing even one undedicated day to the service of Hashem. This cheshbon includes not only positive mitzvos but the fulfillment of all 365 לא תעשה as well as alluded to in the gematriah זקנים באים בימים (including the three words) equals 365.
Avraham said to his guests whom he desired to bring them close to Hashem והשענו תחת העץ to align their שעון their times under the עץ under the control and guidance of the eternal Torah which created time itself. Without Torah scheduling one's spiritual time from the 8th Day of Milah and onwards until death, our lives would never come out as spiritual as they are. The root wordהשענ is the gematriah of כימי העץ ימי עמי . We must make sure that our watches keep the right time always synched with our Torah.