Jambudvipa and the related Jambu tree have been mentioned in various Puranas. The Jambu tree is nothing but the rose apple tree.
Here are a few references from the Devi Bhagavatam which mention about the tree and the Dvipa (that is named after it):
Thus the seven oceans had their origins. And the portion of the earth,
that was included within the ruts, became the seven islands (Dvîpas).
0 Child! Now hear about the seven Dvîpas and the seven Oceans :-- The
first is the Jambu Dvîpa; the second is Plaksa, the third is S'almalî;
the fourth is the Kus'a Dvîpa; the fifth is Krauncha; the sixth is the
S'aka Dvîpa; and the seventh is the Puskara Dvîpa. The second Dvîpa
Plaksa is twice the first Jambu Dvîpa and so on; each succeeding Dvîpa
is twice as large as its previous one. Now hear the names of the
Devi Bhagavatam Book 8; Chapter 4
1-31. S'rî Nârâyana said :-- O Child Nârada! Now hear in detail about
the divisions of the earth into the Dvîpas and the Varsas as marked
out by the Devas. In brief, I describe about them; no one can speak
about this in details. First, the Jambu Dvîpa is one lâkh Yoyanas in
its dimensions. This Jambu Dvîpa is round like a lotus. There are nine
Varsas in it and excepting the Bhadrâs'va and Ketumâla, each is nine
thousand Yoyanas in its dimensions (i. e., in its diameter or
circumference?) and there are eight very lofty mountains, in those
Varsas, forming their boundaries. Of the Varsas, the two Varsas that
are situated in the North and South, are of the size of a bow
(segmental); and the four others are elongated in their size. The
centre of all these Varsas is named Îlâvrita Varsa and its size is
rectangular. In the centre of this Îlâvarsa is situated the golden
Sumeru Mountain, the King of all the mountains, one lakh Yoyanas high.
It forms the pericarp of the lotus earth.
1-37. Nârâyana said :-- The Jambudvîpa has been described, how it is
and what is its width. It is surrounded on all sides by the salt
ocean. As Meru is surrounded by Jambudvîpa, so the salt ocean is
surrounded by Plaksadvîpa, twice its size. As the ditch is surrounded
by gardens, so the salt ocean is surrounded by gardens. As the Jambu
tree exists in Jambudvîpa, so the Plaksa tree exists in the Plaksa
Dvîpa and of the same size. The name Plaksa Dvîpa is derived from the
name of this Plaksa tree. This tree is of a golden colour. Fire exists
at its bottom with form incarnate. This is, named Saptajihva.
Devi Bhagavatam Book 8; Chapter 12
The translator (Swami Vijnanananda) identifies Jambu to be rose apple.
The World Mother becomes very much pleased when one offers to Her the
following things :-- Gur, honey, ghee, milk, curd, Takra, apûpa, fresh
butter, Karkatî, Kusmânda, Modaka, Panasa, plantain, Jambu
(rose-apple), mangoe, sesamum, oranges, Dâdima, (pomegranate) Vadarî
(Jujube) the Dhâtrî (Âmalaki) fruit, Pâyasa, Prithuka, gram, cocoanut,
Jambîra. Kaseru, and S’ûrana.