Solution Explanation: Complete Binary Tree Inserter

This problem involves designing a data structure and algorithms to efficiently insert nodes into a complete binary tree while maintaining its completeness. A complete binary tree is one where all levels are completely filled except possibly the last, and all nodes are as far left as possible.

The solution uses a Breadth-First Search (BFS) approach combined with an array-based representation of the tree. This allows for O(1) insertion and root retrieval.

Algorithm:

1. __init__(root) (Constructor):

   • Initializes a list tree to store all nodes of the tree.
   • Performs a BFS traversal of the input root to populate the tree list. Each node is added to the list in level-order (from left to right, level by level).

2. insert(val):

   • Finds the parent node p of the new node to be inserted. Since the tree is complete, the parent of the last node is always at index (tree.length - 1) // 2.
   • Creates a new node node with the given val.
   • Adds node to the tree list.
   • Attaches node as the left child of p if p.left is None; otherwise, it attaches node as the right child of p.
   • Returns the value of the parent node p.

3. get_root():

   • Returns the root node of the tree, which is always the first element in the tree list (index 0).

Time Complexity Analysis:

• __init__(root): The BFS traversal takes O(N) time, where N is the number of nodes in the tree.
• insert(val): All operations within this function (finding the parent, creating the node, adding to the list, and attaching as a child) take O(1) time.
• get_root(): Retrieving the root node from the list takes O(1) time.

Space Complexity Analysis:

• The tree list stores all nodes of the tree, so the space complexity is O(N).

Code Implementation (Python):

from collections import deque
from typing import Optional
 
# Definition for a binary tree node.
class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right
 
class CBTInserter:
    def __init__(self, root: Optional[TreeNode]):
        self.tree = []
        q = deque([root])
        while q:
            for _ in range(len(q)):
                node = q.popleft()
                self.tree.append(node)
                if node.left:
                    q.append(node.left)
                if node.right:
                    q.append(node.right)
 
    def insert(self, val: int) -> int:
        p = self.tree[(len(self.tree) - 1) // 2]
        node = TreeNode(val)
        self.tree.append(node)
        if p.left is None:
            p.left = node
        else:
            p.right = node
        return p.val
 
    def get_root(self) -> Optional[TreeNode]:
        return self.tree[0]

The implementations in other languages (Java, C++, Go, TypeScript, JavaScript) follow the same algorithmic structure, differing only in syntax and data structure specifics. They all achieve the same time and space complexities.