# 2764. Is Array a Preorder of Some ‌Binary Tree

## Description

Given a 0-indexed integer 2D array nodes, your task is to determine if the given array represents the preorder traversal of some binary tree.

For each index i, nodes[i] = [id, parentId], where id is the id of the node at the index i and parentId is the id of its parent in the tree (if the node has no parent, then parentId == -1).

Return true if the given array represents the preorder traversal of some tree, and false otherwise.

Note: the preorder traversal of a tree is a recursive way to traverse a tree in which we first visit the current node, then we do the preorder traversal for the left child, and finally, we do it for the right child.

Example 1:

Input: nodes = [[0,-1],[1,0],[2,0],[3,2],[4,2]]
Output: true
Explanation: The given nodes make the tree in the picture below.
We can show that this is the preorder traversal of the tree, first we visit node 0, then we do the preorder traversal of the right child which is [1], then we do the preorder traversal of the left child which is [2,3,4].


Example 2:

Input: nodes = [[0,-1],[1,0],[2,0],[3,1],[4,1]]
Output: false
Explanation: The given nodes make the tree in the picture below.
For the preorder traversal, first we visit node 0, then we do the preorder traversal of the right child which is [1,3,4], but we can see that in the given order, 2 comes between 1 and 3, so, it's not the preorder traversal of the tree.


Constraints:

• 1 <= nodes.length <= 105
• nodes[i].length == 2
• 0 <= nodes[i][0] <= 105
• -1 <= nodes[i][1] <= 105
• The input is generated such that nodes make a binary tree.

## Solutions

Solution 1：Depth-First Search

First, we construct a graph $g$ based on the $nodes$ data, where $g[i]$ represents all the child nodes of node $i$.

Next, we design a function $dfs(i)$, which represents a pre-order traversal starting from node $i$. We use a variable $k$ to represent the $k$-th node in the $nodes$ list that we have currently traversed, with an initial value of $k = 0$.

The execution logic of the function $dfs(i)$ is as follows:

If $i \neq nodes[k][0]$, it indicates that the current sequence is not a pre-order traversal sequence of a binary tree, and returns false. Otherwise, we increment $k$ by $1$, and then recursively search all child nodes of $i$. If a false is found during the search, we return false immediately. Otherwise, when the search is finished, we return true. In the main function, we call $dfs(nodes[0][0])$. If the return value is true and $k = |nodes|$, then the $nodes$ sequence is a pre-order traversal sequence of a binary tree, and we return true; otherwise, we return false.

The time complexity is $O(n)$ and the space complexity is $O(n)$, where $n$ is the number of nodes in $nodes$.

• class Solution {
private Map<Integer, List<Integer>> g = new HashMap<>();
private List<List<Integer>> nodes;
private int k;

public boolean isPreorder(List<List<Integer>> nodes) {
this.nodes = nodes;
for (var node : nodes) {
}
return dfs(nodes.get(0).get(0)) && k == nodes.size();
}

private boolean dfs(int i) {
if (i != nodes.get(k).get(0)) {
return false;
}
++k;
for (int j : g.getOrDefault(i, List.of())) {
if (!dfs(j)) {
return false;
}
}
return true;
}
}

• class Solution {
public:
bool isPreorder(vector<vector<int>>& nodes) {
int k = 0;
unordered_map<int, vector<int>> g;
for (auto& node : nodes) {
g[node[1]].push_back(node[0]);
}
function<bool(int)> dfs = [&](int i) {
if (i != nodes[k][0]) {
return false;
}
++k;
for (int j : g[i]) {
if (!dfs(j)) {
return false;
}
}
return true;
};
return dfs(nodes[0][0]) && k == nodes.size();
}
};

• class Solution:
def isPreorder(self, nodes: List[List[int]]) -> bool:
def dfs(i: int) -> int:
nonlocal k
if i != nodes[k][0]:
return False
k += 1
return all(dfs(j) for j in g[i])

g = defaultdict(list)
for i, p in nodes:
g[p].append(i)
k = 0
return dfs(nodes[0][0]) and k == len(nodes)


• func isPreorder(nodes [][]int) bool {
k := 0
g := map[int][]int{}
for _, node := range nodes {
g[node[1]] = append(g[node[1]], node[0])
}
var dfs func(int) bool
dfs = func(i int) bool {
if i != nodes[k][0] {
return false
}
k++
for _, j := range g[i] {
if !dfs(j) {
return false
}
}
return true
}
return dfs(nodes[0][0]) && k == len(nodes)
}

• function isPreorder(nodes: number[][]): boolean {
let k = 0;
const g: Map<number, number[]> = new Map();
for (const [i, p] of nodes) {
if (!g.has(p)) {
g.set(p, []);
}
g.get(p)!.push(i);
}
const dfs = (i: number): boolean => {
if (i !== nodes[k][0]) {
return false;
}
++k;
for (const j of g.get(i) ?? []) {
if (!dfs(j)) {
return false;
}
}
return true;
};
return dfs(nodes[0][0]) && k === nodes.length;
}