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Problem Statement

One way to serialize a binary tree is to use pre-order traversal. When we encounter a non-null node, we record the node’s value. If it is a null node, we record using a sentinel value such as #.

For example, the following binary tree

    9
   / \
  3   2
 / \ / \
4  1 6  #

can be serialized to the string "9,3,4,#,#,1,#,#,2,6,#,#"

Given a string of comma-separated values, verify whether it is a correct preorder traversal serialization of a binary tree. Assumes each comma-separated value can only be an integer or a character '#'. Valid input contains no spaces.

Example 1:

Input: preorder = "9,3,4,#,#,1,#,#,2,#,6,#,#"
Output: true

Example 2:

Input: preorder = "1,#"
Output: false

Example 3:

Input: preorder = "9,#,#,1"
Output: false

Clarifying Questions

1. Is it possible to have a completely empty tree serialized string? No, as the problem states all input strings are valid serializations of a binary tree, not empty.

2. What kind of values can nodes take? Nodes can either be integers or the sentinel value #.

3. Is there any constraint on the length of the input string? Typically, constraints would be provided in the broader problem description, but for this problem, we’ll assume a reasonable length.

Strategy

The goal is to check the validity of a preorder serialized string of a binary tree. Here is a step-by-step strategy:

1. Initialization: Initialize slots to 1. This represents the number of slots available to place nodes. Initially, we have one slot — the root.

2. Iterate through Nodes: Split the input string by commas and iterate through the elements:

   • Processing a node:
     • Decrease the number of slots by one for the current node since it is consuming one slot.
     • If the number of slots becomes negative at any point, the string is invalid.
     • If the node is not #, it will create two additional slots (left and right children).

3. Final Check: After processing all nodes, we should have exactly zero slots left. If we have extra slots, the preorder string is not valid.

Code

Here’s a Python implementation based on the above strategy:

def isValidSerialization(preorder: str) -> bool:
    nodes = preorder.split(',')
    slots = 1  # we begin with one slot available (the root)

    for node in nodes:
        slots -= 1  # each node needs one slot
        
        if slots < 0:
            return False
        
        if node != '#':
            slots += 2  # non-null node generates two additional slots
        
    return slots == 0

# Example usage:
preorder = "9,3,4,#,#,1,#,#,2,#,6,#,#"
print(isValidSerialization(preorder))  # Output: True

Time Complexity

• Time Complexity: O(N), where N is the number of nodes (or the length of the input string split by commas).

  This is because we are iterating through each node exactly once.

• Space Complexity: O(N), due to the space used for storing split nodes. However, we do not use additional space proportional to tree size beyond the splitting process.

Feel free to ask for any further clarifications or additional test cases!

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