Understanding the Problem
A Binary Search Tree (BST) can become unbalanced if new nodes are inserted in sorted order or other skewed sequences. An unbalanced BST results in inefficient operations like search, insert, and delete, as it behaves more like a linked list with O(n) complexity. Our goal is to transform such an unbalanced BST into a balanced one — where the left and right subtrees of every node have roughly the same height. This ensures optimal time complexity for operations.
To do this, we use the property of BSTs that inorder traversal gives a sorted list of values. Once we have that list, we can rebuild a balanced BST by recursively selecting the middle element as root and constructing left and right subtrees from the left and right halves.
Step-by-Step Solution with Example
step 1: Perform Inorder Traversal
Given an unbalanced BST, the first step is to traverse it in inorder (left → root → right). For example, consider this tree:
1
2
3
4
5
Performing inorder traversal on this tree results in a sorted list: [1, 2, 3, 4, 5]
step 2: Build Balanced BST from Sorted List
The idea is to pick the middle element of the list as the root. For [1, 2, 3, 4, 5]
, the middle is 3
. We make 3 the root.
- Left half [1, 2]
becomes the left subtree.
- Right half [4, 5]
becomes the right subtree.
Repeat the same process recursively:
- Middle of
[1, 2]
is 2
→ left child of 3
- Middle of
[4, 5]
is 4
→ right child of 3
Final tree:
3
/ 2 4
/ 1 5
This tree is balanced — the height difference between left and right subtrees at any node is minimal.
step 3: Use Recursion to Construct Subtrees
The building process is recursive:
- If the current list is empty, return null.
- Find the middle element and make it the current root.
- Recursively repeat for left and right halves.
Edge Cases
Empty Tree
If the input BST is empty (null root), there is nothing to balance. The algorithm should simply return null without errors.
Single Node Tree
A tree with just one node is already balanced. The algorithm will return the node as-is, as there are no children to reorganize.
Already Balanced Tree
If the BST is already balanced (e.g., built correctly with middle insertion logic), the algorithm will reconstruct it in the same balanced structure. This shows the solution is idempotent and safe to run multiple times.
Highly Skewed Tree
Trees where all nodes are to one side (left or right) — such as the example above — benefit the most. Balancing converts them from height O(n) to height O(log n).
Finally
This approach to balancing a BST is intuitive, efficient, and optimal for most real-world scenarios. It leverages the BST's sorted property and uses recursion to ensure that every subtree is built with a minimal height difference. This technique is widely used in self-balancing tree structures like AVL Trees and Red-Black Trees (though with additional constraints).
In summary, balance your BST by:
- Collecting nodes using inorder traversal (sorted list)
- Building a new BST using the middle element as root recursively
This guarantees efficient operation performance and maintains the original BST semantics.
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