3575 - Maximum Good Subtree Score.

Hard

You are given an undirected tree rooted at node 0 with n nodes numbered from 0 to n - 1. Each node i has an integer value vals[i], and its parent is given by par[i].

A subset of nodes within the subtree of a node is called good if every digit from 0 to 9 appears at most once in the decimal representation of the values of the selected nodes.

The score of a good subset is the sum of the values of its nodes.

Define an array maxScore of length n, where maxScore[u] represents the maximum possible sum of values of a good subset of nodes that belong to the subtree rooted at node u, including u itself and all its descendants.

Return the sum of all values in maxScore.

Since the answer may be large, return it modulo <code>10⁹ + 7</code>.

Example 1:

Input: vals = 2,3, par = -1,0

Output: 8

Explanation:

• The subtree rooted at node 0 includes nodes {0, 1}. The subset {2, 3} is good as the digits 2 and 3 appear only once. The score of this subset is 2 + 3 = 5.

• The subtree rooted at node 1 includes only node {1}. The subset {3} is good. The score of this subset is 3.

• The maxScore array is [5, 3], and the sum of all values in maxScore is 5 + 3 = 8. Thus, the answer is 8.

Example 2:

Input: vals = 1,5,2, par = -1,0,0

Output: 15

Explanation:

• The subtree rooted at node 0 includes nodes {0, 1, 2}. The subset {1, 5, 2} is good as the digits 1, 5 and 2 appear only once. The score of this subset is 1 + 5 + 2 = 8.

• The subtree rooted at node 1 includes only node {1}. The subset {5} is good. The score of this subset is 5.

• The subtree rooted at node 2 includes only node {2}. The subset {2} is good. The score of this subset is 2.

• The maxScore array is [8, 5, 2], and the sum of all values in maxScore is 8 + 5 + 2 = 15. Thus, the answer is 15.

Example 3:

Input: vals = 34,1,2, par = -1,0,1

Output: 42

Explanation:

• The subtree rooted at node 0 includes nodes {0, 1, 2}. The subset {34, 1, 2} is good as the digits 3, 4, 1 and 2 appear only once. The score of this subset is 34 + 1 + 2 = 37.

• The subtree rooted at node 1 includes node {1, 2}. The subset {1, 2} is good as the digits 1 and 2 appear only once. The score of this subset is 1 + 2 = 3.

• The subtree rooted at node 2 includes only node {2}. The subset {2} is good. The score of this subset is 2.

• The maxScore array is [37, 3, 2], and the sum of all values in maxScore is 37 + 3 + 2 = 42. Thus, the answer is 42.

Example 4:

Input: vals = 3,22,5, par = -1,0,1

Output: 18

Explanation:

• The subtree rooted at node 0 includes nodes {0, 1, 2}. The subset {3, 22, 5} is not good, as digit 2 appears twice. Therefore, the subset {3, 5} is valid. The score of this subset is 3 + 5 = 8.

• The subtree rooted at node 1 includes nodes {1, 2}. The subset {22, 5} is not good, as digit 2 appears twice. Therefore, the subset {5} is valid. The score of this subset is 5.

• The subtree rooted at node 2 includes {2}. The subset {5} is good. The score of this subset is 5.

• The maxScore array is [8, 5, 5], and the sum of all values in maxScore is 8 + 5 + 5 = 18. Thus, the answer is 18.

Constraints:

• 1 <= n == vals.length <= 500

• <code>1 <= valsi<= 10⁹</code>

• par.length == n

• par[0] == -1

• 0 <= par[i] < n for i in [1, n - 1]

• The input is generated such that the parent array par represents a valid tree.