Time complexity of insertion and removing in 2-3 trees

Question

Why do insertion and removing operations in 2-3 tree always have complexity of O(logn), is there a mathematical proof?

I know it's O(logn) and understand why. But I don't understand why removing and insertion operations have the same complexity — Аяз Хуснутдинов, May 19 '18 at 18:26
But what about split operation when inserting a value to a node? It can be performed all the way up to the root. — Аяз Хуснутдинов, May 19 '18 at 18:33
You write about subtree below the place I have to insert. But I can insert only in leaves. What subtree are writing about? — Аяз Хуснутдинов, May 19 '18 at 19:19

score 1 · Accepted Answer · answered May 19 '18 at 19:29

When we insert a key at level , in the worst case we need to split + 1 nodes (one at each of the levels plus the root).
A 2-3 tree containing keys with the maximum number of levels takes the form of a binary tree where each internal node has one key and two children.
In such a tree = (2^(+1)) − 1 where is the number of the lowest level.
This implies that + 1 = log( + 1) from which we see that the splits are in the worst case log .
So insertion in a 2-3 tree takes at worst time.
Similarly we can prove that searches and deletions take time.

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