Returning to Abraham I find Chapter 18 a little bit confusing. Let's see why. In verses 1 through 6 we find the encounter Abraham has with the Lord near the great trees of Mamre. The questions that pop into my head are: 1) How does Abraham so easily recognize the Lord?, and 2) What are the "great trees of Mamre"? I can't really answer the first question. The passage says (verse 2) "Abraham looked up and saw three men standing nearby." and it isn't quite clear if this is the Lord and two men or a separate event of three men just after the Lord has appeared to Abraham. I suppose it could be either. The passage doesn't make it at all clear which. The number three of course, due to later associations, puts me in mind of the Trinity. So are these three all manifestations of the Lord, or are they the Lord and two angels, or is the Lord separate and there are three men or angels associated with the Lord. Since this passage is contiguous with the story of Sodom and Gomorrah we will see more of these "men."
In wondering about the significance of the "trees of Mamre" I did some googling around which is my wont. I am not a deep researcher when I'm thinking about these things. I'm not a biblical scholar. I'm a techie geek scientist who loves writing and poetry and metaphor but not literature research so much. But I did stumble upon this reflection on the tree of Mamre which I think worth reading. It isn't clear why the trees of Mamre are mentioned except perhaps as a land mark. As I read various posts I found that the tree believed to be related at least as a descendant of the tree of Mamre was an oak that is now dead that is itself only some 300 years (another site says it is over 5000 years old) old and so not, of course, the original tree of Mamre. Perhaps it is just that trees are rare in the desert that made it a landmark. There seems to also be traditions linked to the tree: "A long-standing tradition is that the Oak of Abraham will die before the appearance of the Anti-Christ. The oak has been dead since 1996." So there are confusing elements of story about the symbol of the tree.
Verse three tends to make me more comfortable with the idea that there may be a Trinitarian link here — but one has to be careful with translations. Still notice that Abraham extends hospitality of water to wash their feet and rest and goes (verse 6) to have Sarah bake bread and then has a servant kill a choice tender calf. So we have some rather extravagant hospitality and to the request that the Lord stay, the verse says "they (i.e. collectively) answered" which I suppose is a small point, but tends to support the idea that "Lord" is being applied collectively.