We are in the Jyana section and in this section, Krishna is systematically establishing how Jyana culminates in Bhakti. 

In the 13th chapter, He talked about 6 items: Purusha, Prakriti, Jyana, Jneya, Kshetra, Kshetra-jyna and based on that He explained how by Bhakti one can get a proper understanding of these. In the 14th chapter, it is further elaborated through the analysis of material nature, by which one understands that if one transcends the modes one has to take shelter of devotional service. And that devotional service culminates in understanding Krishna is the support of Brahman. Now after concluding the 14th chapter and emphasizing that Bhagavan is superior to Brahman, Krishna then elaborates on that theme differently. He explains Bhagavan as the ultimate reality by emphasizing that Bhagavan is Purushotama the Greatest Person. So this chapter is called ‘Purushotam Yoga’ translated as ‘The Yoga of the Supreme Person’ by Srila Prabhupada

Chapter 15 Overview

1-6: Attain Liberation by detachment (The upside-down banyan tree)

7-11: Otherwise there will be bondage and transmigration

12-15: How Krishna assists us in our material existence and our spiritual endeavor

16-18: The essence of Vedanta (Tri-Shloki Gita)

19-20: To know Krishna means to know everything

Bg 15.1

śrī-bhagavān uvāca
ūrdhva-mūlam adhaḥ-śākham
aśvatthaṁ prāhur avyayam
chandāṁsi yasya parṇāni
yas taṁ veda sa veda-vit

Synonyms: 

śrī–bhagavān uvāca — the Supreme Personality of Godhead said; ūrdhva–mūlam — with roots above; adhaḥ — downwards; śākham — branches; aśvattham — a banyan tree; prāhuḥ — is said; avyayam — eternal; chandāṁsi— the Vedic hymns; yasya — of which; parṇāni — the leaves; yaḥ — anyone who; tam — that; veda — knows; saḥ — he; veda–vit — the knower of the Vedas.

Translation: 

The Supreme Personality of Godhead said: It is said that there is an imperishable banyan tree that has its roots upward and its branches down and whose leaves are the Vedic hymns. One who knows this tree is the knower of the Vedas.

ūrdhva-mūlam adhaḥ-śākham

ūrdhva-mūlam, that which is above

adhaḥ-śākham So the roots are above and branches are below

aśvatthaṁ prāhur avyayam

aśvattham – a banyan tree

prāhur avyayam- it is said to be imperishable

chandāṁsi yasya parṇāni-parṇāni are its leaves. So its leaves are the chanda, the Vedic hymns

yas taṁ veda, one who knows this tree in totality is the sa veda-vit, the knower of the Veda

Significantly, Krishna says that the leaves are the Vedic Hymns. Just knowing the leaves alone is not enough, by just knowing Vedic Hymns one doesn’t become veda-vit, to become Veda-vit one has to know the complete tree. 

In 2.53 Krishna speaks about Trai Gunya Vishya Veda and tells Arjuna to go beyond the three modes. Similarly, in 6.43-44, He has talked earlier about shabda brahma ati vartate: when the transcendentalist who is going on a multi-lifetime journey comes close to perfection s/he is attracted spontaneously by the principles of spirituality and goes beyond Karma Kanda i.e. the range of fruitive activities which are talked about in the Vedic literature. 

So one who is the knower of the Vedas, a ved-vit should not just know the Vedic Hymns, they are only the leaves within the tree. Leaves are just one part of the tree but along with the leaves there are other parts that the tree is comprised of. And to understand the tree, its nature and purpose, one who knows that is the knower of the Vedas. 

So what is this tree being referred to here? One thing is very clear that it is a metaphor. This tree is used to refer to material existence. When we talk about material existence we are talking about the soul’s existence within matter. Matter is always existing but matter is not aware of its existence. So when we talk about material existence, that is the soul’s existence within matter. The tree gives the example of existence of matter; not just the existence of matter but the existence of the soul within the arena of matter. 

And the tree represents material existence in the sense that here things are upside down, things are opposite of the way they should be. In this material world, Krishna is not in the center, the soul has become the center, this will be talked about in 15.7.

Acharyas have explained that we see the upside-down tree in reflection so this material world is like a reflection of the spiritual world.

As we know material existence itself is imperishable, it is eternal. It may be manifested sometime it may not be manifested sometime. Matter is sometimes present, sometimes absent but it is avyaya. 

Aśvattha has many meanings, one meaning is that which will not last. Material existence is eternal but we are not attracted to material existence per se we are attracted to the forms, we are attracted to the beautiful objects and those are not permanent, those are temporary. Because they are temporary there cannot be lasting happiness in those relationships. Relationships with other people, relationships between things of this material world. Are we attracted to the substratum of matter or to the soul? Generally, we are attracted to outer appearances and those appearances will disappear, they will not last.

Aśvattha has another meaning- Asva is horse, tha- a horse has habit of being situated on three feet, so similarly this world is situated unsteadily, it is eka pad vibhuti, three pad of it is on the spiritual world and we know that all material world is ek pad, it is one-fourth of the totality of existence. The major portion of material existence is in the spiritual world.  

chandāṁsi yasya parṇāni
yas taṁ veda sa veda-vit

So Krishna is introducing a metaphor by pointing out that things are upside down over here. 

SP Purport: For one who is engaged in fruitive activities, there is no end to the banyan tree.

… The fruits represent the results of the living entities’ activities, namely, religion, economic development, sense gratification and liberation.

From Vedic Hymns-Dharma, Artha, Kama & Moksha can come about.

SP Purport: Desire is the cause of things being situated in this reflected material light.

Because we have the desire to enjoy ourselves separately from Krishna that is why we are in the material existence. 

SP Purport: A reflection is temporary, for it is sometimes seen and sometimes not seen. But the origin from whence the reflection is reflected is eternal. The material reflection of the real tree has to be cut off. When it is said that a person knows the Vedas, it is assumed that he knows how to cut off attachment to this material world. If one knows that process, he actually knows the Vedas. One who is attracted by the ritualistic formulas of the Vedas is attracted by the beautiful green leaves of the tree. He does not exactly know the purpose of the Vedas. The purpose of the Vedas, as disclosed by the Personality of Godhead Himself, is to cut down this reflected tree and attain the real tree of the spiritual world.