To solve this problem using the Disjoint Set Union (DSU) approach, we aim to union nodes based on their parent-child relationships. A valid tree should follow these rules:
Time Complexity: O(n), where n is the number of nodes, due to each union and find operation being nearly constant with path compression.
Space Complexity: O(n) for the parent array.
1class Solution:
2 def find(self, parent, x):
3 if parent[x] != x:
4 parent[x] = self.find(parent, parent[x])
5 return parent[x]
6
7 def union(self, parent, x, y):
8 rootX = self.find(parent, x)
9 rootY = self.find(parent, y)
10 if rootX == rootY:
11 return False
12 parent[rootY] = rootX
13 return True
14
15 def validateBinaryTreeNodes(self, n, leftChild, rightChild):
16 parent = list(range(n))
17 hasParent = [0] * n
18
19 for i in range(n):
20 if leftChild[i] != -1:
21 if hasParent[leftChild[i]] or not self.union(parent, i, leftChild[i]):
22 return False
23 hasParent[leftChild[i]] = 1
24 if rightChild[i] != -1:
25 if hasParent[rightChild[i]] or not self.union(parent, i, rightChild[i]):
26 return False
27 hasParent[rightChild[i]] = 1
28
29 return hasParent.count(0) == 1The Python solution follows the union-find pattern, organized into find and union methods. The parent list captures each node’s representative root, while hasParent ensures a node has at most one parent. Disjoint set operations verify no cycles and ensure all nodes unify under one connected root without duplication errors, confirming only one root node.
This approach involves calculating the in-degree of each node and checking connectivity via a DFS. The key aspects of a tree like single-root presence and cycle-checking can be managed by:
Time Complexity: O(n), since each node and its immediate edges are evaluated once in each step, including in-drives calculations and DFS.
Space Complexity: O(n) for holding visited tracking and in-degree counts.
1#include <stdbool.h>
2#include <string.h>
3
4void dfs(int node, int* visited, int** children, int n) {
5 if (visited[node]) return;
6 visited[node] = 1;
7 if (children[0][node] != -1) dfs(children[0][node], visited, children, n);
8 if (children[1][node] != -1) dfs(children[1][node], visited, children, n);
9}
10
11bool validateBinaryTreeNodes(int n, int* leftChild, int leftChildSize, int* rightChild, int rightChildSize) {
12 int inDegree[n];
13 memset(inDegree, 0, sizeof(inDegree));
14
15 for (int i = 0; i < n; ++i) {
16 if (leftChild[i] != -1) ++inDegree[leftChild[i]];
17 if (rightChild[i] != -1) ++inDegree[rightChild[i]];
18 }
19
20 int root = -1;
21 for (int i = 0; i < n; ++i) {
22 if (inDegree[i] == 0) {
23 if (root == -1) {
24 root = i;
25 } else {
26 return false; // more than one root
27 }
28 }
29 }
30
31 if (root == -1) return false; // no root
32
33 int visited[n];
34 memset(visited, 0, sizeof(visited));
35 dfs(root, visited, (int*[]){leftChild, rightChild}, n);
36
37 for (int i = 0; i < n; ++i) {
38 if (!visited[i]) return false; // not fully connected
39 }
40
41 return true;
42}In this C solution, the algorithm starts by calculating the in-degree for each node, which should equal zero for the root node. A DFS checks connectivity from a root node, ensuring all nodes are part of one connected graph and are reachable. Each node is visited once, ensuring an entire traversal defines the single connected component rule of trees.