The tree associated with Bhagavan Sri Krishna is the Kadamba
Tree (Burflower tree - Anthocephalus Cadamba). As per the Srimad Bhagavad Purana, Sri Krishna
climbed on a Kadamba tree after stealing the clothes of gopis. It is also said
in the same Purana, that Krishna played his flute sitting on a Kadamba tree.
Kadamba tree is also worshipped in Varanasi and other
regions in North India.The symbol of the tree is also the symbol of the tree of life – with its abundant and simultaneous flowering and fruiting, and its sheltering of animal life. The roots of the tree are in the nether world in the mound of water, while its crown reaches far up into the sky. The tree thus represents the simultaneity and the interpenetration of human, natural and divine realms. (Source - Book titled Krishna by Shanta Rameshwar Rao)
Sri Krishna and Radha used to meet under the Kadamba Tree. Thus the tree is associated with lovers. He used to swing on the branches of tree and used to dance with the Gopis under it.
It is believed that Sri Krishna jumped from a Kadamba tree into the Yamuna River to fight the snake Kaliya, who was poisoning the waters of Yamuna.
Sap of Kadamba Led to End of Yadava Clan
There is a story which suggest that the Yadava Clan drank the sap of Kadamba Tree and got intoxicated. They insulted Rishis and Munis. The merry making continued for long and finally the clan got cursed and it result in the end of Yadava Clan and departure of Sri Krishna from earth. (You can read the story here)Medicinal Uses of Kadamba
Paste made using Kadamba leaf is applied over wound and
swelling.
A decoction made from the bark of the tree is used to wash
infection related to wood. The same decoction is used for gargling to treat
mouth ulcers and inflammation of the gums.