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Is it possible to implement binary heap without using additional data structure (e.g. array, linked list)

Can I implement a binary heap by only using a TreeNode inferface (has children, or left/right, or/and parent.. something like this)?

I want to not rely on using array or linked list.

If I don't use array or linked list, I have a trouble inserting the next element in the correct place & keep it a complete binary tree (all non-leaf nodes are full). Also have trouble taking out the root and re-heapifying.