Insertion follows the path a search for the value would take, then attaches the new value where that search would have failed. You walk down comparing, going left for smaller and right for larger, until you reach an empty child slot. You place the new node there as a leaf.
New values always land as leaves, so existing nodes never move. That keeps insertion simple and preserves the ordering invariant, since you followed it on the way down.
Duplicates need a rule decided up front, like sending equals right or bumping a count on the existing node.
The cost mirrors search. It is proportional to height, so O(log n) on a balanced tree and O(n) on a skewed one. Sorted input builds a long one-sided chain that every operation walks like a linked list, which is why real systems self-balance after inserts.
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