Merge Two Balanced Binary Search Trees
You are given two balanced binary search trees e.g., AVL or Red-Black Tree. Write a function that merges the two given balanced BSTs into a balanced binary search tree. Let there be m elements in the first tree and n elements in the other tree. Your merge function should take O(m+n) time.
In the following solutions, it is assumed that the sizes of trees are also given as input. If the size is not given, then we can get the size by traversing the tree (See this).
Method 1 (Insert elements of the first tree to second):
Take all elements of first BST one by one, and insert them into the second BST. Inserting an element to a self balancing BST takes Logn time (See this) where n is size of the BST. So time complexity of this method is Log(n) + Log(n+1) … Log(m+n-1). The value of this expression will be between mLogn and mLog(m+n-1). As an optimization, we can pick the smaller tree as first tree.
Method 2 (Merge Inorder Traversals):
- Do inorder traversal of first tree and store the traversal in one temp array arr1. This step takes O(m) time.
- Do inorder traversal of second tree and store the traversal in another temp array arr2. This step takes O(n) time.
- The arrays created in step 1 and 2 are sorted arrays. Merge the two sorted arrays into one array of size m + n. This step takes O(m+n) time.
- Construct a balanced tree from the merged array using the technique discussed in this post. This step takes O(m+n) time.
Time complexity of this method is O(m+n) which is better than method 1. This method takes O(m+n) time even if the input BSTs are not balanced.
Following is implementation of this method.
Following is Inorder traversal of the merged tree 20 40 50 70 80 100 120 300
Method 3 (In-Place Merge using DLL):
We can use a Doubly Linked List to merge trees in place. Following are the steps.
- Convert the given two Binary Search Trees into doubly linked list in place (Refer this post for this step).
- Merge the two sorted Linked Lists (Refer this post for this step).
- Build a Balanced Binary Search Tree from the merged list created in step 2. (Refer this post for this step)
Time complexity of this method is also O(m+n) and this method does conversion in place.
Thanks to Dheeraj and Ronzii for suggesting this method.