Problem Description

The problem provides us with a 2D integer array, nodes, that we need to analyze to determine if it represents a preorder traversal of a binary tree. Preorder traversal is a method of visiting all the nodes in a tree where we first visit the node itself, then the left subtree, and finally the right subtree, recursively.

Each element of nodes is in the form [id, parentId], where id is the identifier for the current node, and parentId is the identifier for this node's parent. If the node is the root of the tree, it will have a parentId of -1.

A valid preorder traversal is one where nodes are visited exactly in the order they are laid out in nodes. The task is to return true if nodes represents such a valid preorder traversal, otherwise return false.

Flowchart Walkthrough

To analyze LeetCode 2764 and determine the appropriate traversal or search technique using the Flowchart, here’s a step-by-step walkthrough:

Is it a graph?

• Yes: As the problem identification of whether an array is a preorder traversal involves understanding tree structure, trees can be considered special cases of graphs.

Is it a tree?

• Yes: The problem specifically concerns a binary tree—it asks whether an array can represent the preorder traversal of some binary tree.

Is the problem related to directed acyclic graphs (DAGs)?

• No: This problem specifically concerns binary trees and their traversal mechanism, not properties or operations typically associated with DAGs like topological sorting.

Conclusion: By following the flowchart from identifying a graph, recognizing it as a tree, and dismissing DAG considerations, it leads directly to the application of a tree-based approach. Since the problem involves checking if an array is a preorder traversal, the Depth-First Search (DFS) pattern is suggested as a method to simulate and check the tree's preorder traversal against the given array.

Intuition

To validate that nodes is the preorder traversal of a binary tree, we can model the traversal process using a Depth-First Search (DFS) approach.

Our main tool is a recursive function dfs(i) that undertakes the traversal process. We use it to traverse nodes as if we're moving through a binary tree in preorder and checking if the nodes array matches this traversal.

Here's the intuition behind each step taken to come up with the solution:

1. The given sequence's element alignment must simulate entering each node, then delving into its child nodes as encountered in preorder traversal.

2. Starting from the first node (which we expect to be the root), we aim to traverse its children in the order given in nodes.

3. We utilize a pointer k to keep track of our position in nodes during traversal.

4. The graph g represents the tree. In g, we store child nodes under their respective parent nodes. If a node doesn't appear as any node's parent, then it would not exist in g which implies that it's either a leaf or an unconnected node (which would result in false).

5. For the DFS function dfs(i), we check if the current node index i matches the index nodes[k][0]. If not, this suggests a mismatch in the traversal order, hence the current nodes cannot represent a preorder traversal sequence.

6. We then increment k by 1 and recursively apply dfs to the children nodes, recorded in g[i].

7. If the DFS goes through all the nodes successfully with the indexes matching the preorder traversal sequence, and k equals the length of nodes (meaning every node in nodes has been visited in sequence), nodes is a valid preorder traversal sequence.

In summary, the solution employs a graph representation and DFS to simulate and verify the preorder traversal, checking whether nodes correspond to such a traversal for some tree. If the preorder traversal process is completed without mismatches and includes all nodes, we confirm that nodes represents a valid preorder traversal.

Learn more about Stack, Tree, Depth-First Search and Binary Tree patterns.

Solution Approach

The solution to this problem leverages depth-first search (DFS) to emulate the preorder traversal of the binary tree that nodes is expected to represent.

Here's a step-by-step walkthrough of how we implement this:

1. Graph Construction: We initialize an empty graph g using defaultdict(list). This graph maps parent nodes to their child nodes. For each node i with a parent p in nodes, we append i to g[p]. Nodes with parentId of -1 won’t have any entry in g as they are the roots.

2. DFS Function (dfs): The recursive function dfs(i) embodies the exploration logic. It's designed to perform the following actions:

   • Check whether the current node index i matches the current node index in nodes[k].
   • If there is a mismatch (i != nodes[k][0]), the function returns False, as the current sequence does not reflect a valid preorder traversal.
   • When a match is found, we increment the global variable k and proceed to recursively apply dfs to each child node listed under g[i] in the graph.

3. Recursive Traversal: The dfs function attempts to visit all child nodes of the current node i recursively, checking for continuity and consistency with nodes. When dfs visits a new node, it looks for a match at the current index k. The process is repeated until all nodes in the graph g have been visited, or a mismatch is detected.

4. Validation: After setting up the graph and the dfs function, we begin the traversal with dfs(nodes[0][0]) since we expect the first entry in nodes to represent the root of the binary tree. The traversal only deems valid if two conditions are met post-traversal:

   • The dfs(nodes[0][0]) function should return True, indicating that the traversal completed without detecting any mismatch.
   • The variable k should equal the length of nodes, ensuring that each node was visited exactly once in the correct preorder sequence.

If both conditions are satisfied, we return True, signaling that the nodes array indeed represents the preorder traversal of a binary tree. Otherwise, we return False.

Through this combination of graph data structure and DFS, the solution mirrors the preorder traversal by visiting nodes in the expected sequence and checking for the integrity of the traversal at each step.

Example Walkthrough

Let's consider a small example to illustrate how the solution approach works. Suppose we are given the following nodes array:

nodes = [[1, -1], [2, 1], [3, 2], [4, 1]]

This array is supposed to represent a binary tree's preorder traversal. Let's walk through our algorithm to check whether it's valid.

1. Graph Construction: We begin by constructing our graph g. By iterating through nodes, we populate g as follows:

   • Since nodes[0][1] is -1, node 1 is the root and will not have any parent entry in g.
   • Node 2 is the child of node 1, so we add 2 to g[1].
   • Node 3 is the child of node 2, so we add 3 to g[2].
   • Node 4 is another child of node 1, so we add 4 to g[1].

   The resulting graph g looks like this: {1: [2, 4], 2: [3]}.

2. DFS Function (dfs): Our dfs function will check the sequence of the nodes. We also have a global pointer k initialized to 0:

3. Recursive Traversal: We start our traversal with dfs(nodes[0][0]) which is dfs(1). Inside this function:

   • We check if nodes[k][0] is 1. It is, so we continue.
   • We increment k to 1 and apply dfs to g[1], which contains [2, 4].
   • First, we call dfs(2). We check if nodes[k][0] is 2. It is, so we continue.
   • We increment k to 2 and call dfs(3) since 3 is a child of 2.
   • Inside dfs(3), we check if nodes[k][0] is 3. It is, so we continue.
   • We increment k to 3 and since 3 has no child, dfs(3) returns True.
   • We return to dfs(2) and since there are no more children, it returns True.
   • Back in dfs(1), we proceed with the next child in g[1], which is 4.
   • We call dfs(4) and check if nodes[k][0] is 4. It is, so we continue.
   • We increment k to 4, and since node 4 has no children, dfs(4) returns True.
   • Finally, dfs(1) has no more children to process and returns True.

4. Validation: After the recursive traversal, we check our conditions for a valid preorder traversal:

   • The dfs(nodes[0][0]) function returned True, indicating no mismatches were found.
   • The global counter k is 4, which is equal to the length of nodes, ensuring all nodes were visited in the correct sequence.

Since both conditions are satisfied, we return True. The nodes array [1, -1], [2, 1], [3, 2], [4, 1] does represent a valid preorder traversal of a binary tree.

Solution Implementation

Time and Space Complexity

The time complexity of the solution is O(n), where n is the number of nodes in the nodes list. This is because the algorithm visits each node exactly once due to the dfs traversal, and the k index ensures that we do not revisit nodes. Each call to dfs increments k once and since there are n nodes, we get a linear time complexity.

The space complexity is also O(n). The space is used by the g dictionary, which stores a list of children for each node, and the recursive call stack of the dfs function. In the worst case, if the tree is completely unbalanced, the depth of the recursion can be n, which would mean n function calls on the call stack. Every node also appears in the dictionary, thus, the space complexity is proportional to the number of nodes, giving us O(n) space complexity.

Learn more about how to find time and space complexity quickly using problem constraints.