The Medrash Says, In the merit that Avrohom told the angels, “rest under the tree,” his children merited the Mitzvah of Sukkah.
Now let us explain on what is known that the Eitz Hachayim (tree of life) was in the middle of the Garden.
Now the Eitz Hachayim refers to the Name Haveyeh (inner name of God.)
Now the idea that the Eitz Hachayim was in the Garden, is that the name Haveyeh is the inner part of the 53 (Gan) Parshiyious – Torah portions.
The name of God enclothes itself in many coverings – infinite.
To begin with it clothes itself in the five ways of articulating letters in the mouth.
And for there the letters are formed.
And then the combinations, namely the words.
And then the stories (the Torah of-course is stories.)
Now the Talmud says, whoever has eyes sees the inner side of the king (in the Torah.)
But one lacking eyes peers at the outer.
Now Avrohom thought the angels were people and he taught everyone, so his message to them was, when you are doing something peer not on the outer part or the stories, but the inner.
This is the reason it is the “Tree of life” for there is the source of all blessing and life.
And this is what it says “Man is like a tree of the field” – a composite of body and spirit – and he was saying: ensure that the Neshoma, your Divine soul, is always with you.
Now it is known regarding a Mitzvah that there is man, speech, and action.
And Avrohom Avinu was telling them to “watch my behavior” for when he did a Mitzvah, he placed his soul in it, and he included all three levels.
Now man’s speach and actions, it is known, that action is a branch off of speach and a speach is a branch of thought, and man too is branch of thought – Choshav Ma”h (God’s thought.)