The Tree of Jiva and Atman appears in the Vedic scriptures, predating current Hinduism, as a metaphysical metaphor concerning the soul.
The Rig Veda samhita 1.164.20-22, Mundaka Upanishad 3.1.1-2, and Svetasvatara Upanisad 4.6-7, speak of two birds, one perched on the branch of the tree, which signifies the body, and eating its fruit, the other merely watching.
Rig Veda samhita says:
1.164.20 Two birds associated together, and mutual friends, take refuge in the same tree; one of them eats the sweet fig; the other abstaining from food, merely looks on.
1.164.21 Where the smooth-gliding rays, cognizant, distil the perpetual portion of water; there has the Lord and steadfast protector all beings accepted me, though immature in wisdom.
1.164.22 In the tree into which the smooth-gliding rays feeders on the sweet, enters, and again bring forth light over all, they have called the fruit sweet, but he partakes not of it who knows not the protector of the universe.
The first bird represents a Jiva, or individual self, or soul. She has a female nature, being a shakti, an energy of God. When the jiva becomes distracted by the fruits (signifying sensual pleasure), she momentarily forgets her lord and lover and tries to enjoy the fruit independently of him. This separating forgetfulness is maha-maya, or enthrallment, spiritual death, and constitutes the fall of the jiva into the world of material birth, death, disease and old age.
The second bird is the Paramatman, an aspect of God who accompanies every living being in the heart while she remains in the material world. He is the support of all beings and is beyond sensual pleasure.
It can be stated that this concept of Atman and Jiva have been personified and taken into the Bible as Adam and Eve and the fall of man. It has been opined by a highly revered Shankaracharya of the Kanchi Mutt, His Holiness Chandrashekharendra Saraswati, that the term "Atman" also known commonly as "Atma" could have become "Adam" and "Jeeva", "Eve". Conversely it can also be stated that this abstraction of Jiva and Atman is immutable essence related to the events surrounding the fall of man.
Another interpretation is the two birds are two options: one is to eat the sweet fruits; other is merely look on and recognize where the smooth-gliding rays distil the perpetual water (the leaves and the root); there the lord and protector accepts me in the tree, into which the smooth-gliding rays feeders on the sweet and brings light over the tree. First consciousness has to know the protector of the universe and then partakes of it.
The text itself, however, relates more to the bounties of the world and praise for the creator. The myriad of interpretations of this particular text are seeded more in later Hindu philosophizations influenced over time by the interactions of Hinduism with other philosophies and religions, rather than having any clear historical association with the intended meaning of the specific verse.