Which country is the natural home of tree kangaroos?

It depends on the species.

Bennett's Tree Kangaroo and Lumholtz's Tree Kangaroo both live in the rainforests of Cape York in Far North Queensland, Australia.

There are also numerous species of tree kangaroo on the island of New Guinea, which comprises the countries of Papua New Guinea and Irian Jaya (Indonesia). These tree kangaroos are commonly found in the mountainous rainforests of the island. Several species live in lowlands, for example, the Lowlands Tree-Kangaroo.