How is the insertion operation's time complexity evaluated in a TreeMap data structure?

In Java, the TreeMap data structure is implemented based on a red-black tree — a balanced binary search tree.

The time complexity of inserting an element into a TreeMap is O(log n), where n is the number of elements in the tree. This is because inserting requires finding the correct spot in the tree (logarithmic time) and, if necessary, performing tree balancing, which also takes logarithmic time.

Example:

TreeMap<Integer, String> map = new TreeMap<>();
map.put(10, "ten"); // insertion in O(log n)
map.put(5, "five");
map.put(20, "twenty");

Thus, TreeMap provides efficient insertion, deletion, and search operations with logarithmic complexity.