**π‘ Problem Formulation:** A balanced binary tree is one where the depth of all leaf nodes or nodes with two children differs by no more than one level. This equilibrium is crucial for maintaining optimal performance during operations such as search, insert, and delete. We aim to transform a given binary tree into a balanced one so that these operations can run efficiently. An example input could be a skewed binary tree, and the expected output would be a balanced version of that tree.

## Method 1: Rebuilding the Tree

This method involves creating an in-order list of all nodes in the tree and then rebuilding the tree from that list. A balanced tree is built by recursively finding the middle element of the list and making it the root, then applying the same procedure for sublists. The function specification is as follows: given the root of a binary tree, the method returns the root of a balanced binary tree.

Here’s an example:

def buildTree(self, nums): if not nums: return None mid = len(nums) // 2 node = TreeNode(nums[mid]) node.left = self.buildTree(nums[:mid]) node.right = self.buildTree(nums[mid+1:]) return node def balanceBST(self, root): def inorder(r): return inorder(r.left) + [r] + inorder(r.right) if r else [] return self.buildTree([node.val for node in inorder(root)])

The output would be the new tree root, which now represents a balanced binary tree.

The code snippet functions by first performing an in-order traversal of the given tree to obtain a sorted list of values. This list is then utilized to build a balanced tree as described, ensuring the conditions for a balanced binary tree are met.

## Method 2: AVL Trees

AVL trees are self-balancing binary search trees where the difference between heights of left and right subtrees cannot be more than one for all nodes. Insertions and deletions are accompanied by rotations to ensure the balance is preserved. Through tree rotation operations, the tree is restructured without destroying the binary search tree property.

Here’s an example:

class TreeNode(object): def __init__(self, val): self.val = val self.left = self.right = None self.height = 1 class AVLTree(object): def insert(self, root, key): # Insert code for AVL insertion and call # utility functions to rebalance the tree # This is simplified and not full implementation. def treeInsert(root, key): tree = AVLTree() root = tree.insert(root, key) return root

The output would be the root of the AVL tree with the new key inserted, maintaining a balanced state.

This code snippet represents a small part of an AVL tree insertion operation focusing on the insertion of a new key into an AVL Tree. The full implementation would include additional utility functions to update heights and balance the tree through rotations during insertions and deletions.

## Method 3: Red-Black Trees

Red-Black Trees are another form of self-balancing binary search trees. Each node contains an extra bit for denoting the color of the node, red or black, which ensures the tree remains approximately balanced during insertions and deletions. The algorithm ensures that the properties of Red-Black Trees are preserved every time the tree is modified.

Here’s an example:

class RedBlackTree: # A node has color, value, left and right children, and a parent # Implementation includes leftRotate, rightRotate, insert, insertFixup # And other utility functions to maintain balance after modifications

The output would be the root of a Red-Black Tree maintaining a balanced state after the new insertion.

This code snippet does not provide specific code but outlines that a Red-Black Tree implementation includes various utility functions for maintaining the tree’s balanced state, with particular focus on color attributes and rotations.

## Method 4: Scapegoat Trees

Scapegoat trees are a form of self-balancing binary search tree that maintains balance through a concept called the “scapegoat” node. A node is identified as a scapegoat for rebalancing if its size gets too large compared to its parent. Balancing is achieved by rebuilding the subtree rooted with the scapegoat. This method is less often used compared to AVL or Red-Black Trees but can be more efficient for certain use cases.

Here’s an example:

class ScapegoatTree: # Class implementation with insert, findScapegoat, rebuildTree # Inserting might identify a scapegoat node and then call # rebuildTree to balance the tree again

The output would be the new root of a Scapegoat Tree that has been rebalanced using the scapegoat node.

This synopsis stresses that a Scapegoat Tree uses a specific balancing strategy that involves frequent full subtree rebuilding for maintaining overall balance, which can be effective when insertions are infrequent or batched.

## Bonus One-Liner Method 5: Tree Libraries

For Python developers, the simplest and probably the most effective way for dealing with balanced binary trees is to utilize pre-built libraries that include robust implementations of balanced trees. Libraries such as `bintrees`

or `sortedcontainers`

can provide out-of-the-box support for balanced binary trees without rewriting complex tree algorithms.

Here’s an example:

from sortedcontainers import SortedDict sd = SortedDict() sd.update({key: value}) # Automatically maintains a balanced tree structure

The output is a SortedDict that functions like a balanced binary tree with key-value pairs.

This code uses the SortedDict class from the sortedcontainers library, which internally maintains a balanced tree structure, avoiding the overhead of manual tree balancing implementations.

## Summary/Discussion

**Method 1:**Rebuilding the Tree. Strengths: Conceptually simple and ensures perfect balance. Weaknesses: Not efficient for dynamic or frequently updated trees.**Method 2:**AVL Trees. Strengths: Highly balanced, good for the lookup-intensive workload. Weaknesses: Rotations during insertions/deletions can be complex.**Method 3:**Red-Black Trees. Strengths: Consistently balanced, good for real-time applications. Weaknesses: More complex to implement and understand than AVL trees.**Method 4:**Scapegoat Trees. Strengths: No need for rotation operations. Weaknesses: Tree rebuilding can be expensive for frequent updates.**Bonus Method 5:**Tree Libraries. Strengths: Simple to use, battle-tested. Weaknesses: Less flexibility than custom implementations.