So, if we have n nodes in the AVL tree and we add one more number it will take O(logn) time to insert it. AVL trees balanced binary tree so at any point of time AVL tree will have a height of O(h) (where h=logn).
When we insert any node to the AVL tree following are the steps to be taken.
1. Find the appropriate place of insertion .(i.e do a binary search to get the location to insert element, as it is already sorted it will take O(logn) time.
2. Updating the height (it takes constant time )
3. Rotations (it also takes constant time, as we have to just update some pointers)
Now, if we want to add m more nodes to the tree it will take another m insertions and time complexity for each insertion would be:-
$log(n+1) + log(n+2)+ log(n+3)+ .....+log(n+m)$
=$log((n+1)*(n+2)*(n+3)*....* (n+m-1)*(n+m))$
=$log ( (n+m)! /(n!))$
=$(n+m) log(n+m)- nlogn$
=$O((n+m) log (n+m))$
