Maximize Sum of Weights after Edge Removals
<p>There exists an <strong>undirected</strong> tree with <code>n</code> nodes numbered <code>0</code> to <code>n - 1</code>. You are given a 2D integer array <code>edges</code> of length <code>n - 1</code>, where <code>edges[i] = [u<sub>i</sub>, v<sub>i</sub>, w<sub>i</sub>]</code> indicates that there is an edge between nodes <code>u<sub>i</sub></code> and <code>v<sub>i</sub></code> with weight <code>w<sub>i</sub></code> in the tree.</p>

<p>Your task is to remove <em>zero or more</em> edges such that:</p>

<ul>
	<li>Each node has an edge with <strong>at most</strong> <code>k</code> other nodes, where <code>k</code> is given.</li>
	<li>The sum of the weights of the remaining edges is <strong>maximized</strong>.</li>
</ul>

<p>Return the <strong>maximum </strong>possible sum of weights for the remaining edges after making the necessary removals.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">edges = [[0,1,4],[0,2,2],[2,3,12],[2,4,6]], k = 2</span></p>

<p><strong>Output:</strong> <span class="example-io">22</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/3300-3399/3367.Maximize%20Sum%20of%20Weights%20after%20Edge%20Removals/images/test1drawio.png" style="width: 250px; height: 250px;" /></p>

<ul>
	<li>Node 2 has edges with 3 other nodes. We remove the edge <code>[0, 2, 2]</code>, ensuring that no node has edges with more than <code>k = 2</code> nodes.</li>
	<li>The sum of weights is 22, and we can&#39;t achieve a greater sum. Thus, the answer is 22.</li>
</ul>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">edges = [[0,1,5],[1,2,10],[0,3,15],[3,4,20],[3,5,5],[0,6,10]], k = 3</span></p>

<p><strong>Output:</strong> <span class="example-io">65</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>Since no node has edges connecting it to more than <code>k = 3</code> nodes, we don&#39;t remove any edges.</li>
	<li>The sum of weights is 65. Thus, the answer is 65.</li>
</ul>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= k &lt;= n - 1</code></li>
	<li><code>edges.length == n - 1</code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code><font face="monospace">0 &lt;= edges[i][0] &lt;= n - 1</font></code></li>
	<li><code><font face="monospace">0 &lt;= edges[i][1] &lt;= n - 1</font></code></li>
	<li><code><font face="monospace">1 &lt;= edges[i][2] &lt;= 10<sup>6</sup></font></code></li>
	<li>The input is generated such that <code>edges</code> form a valid tree.</li>
</ul>
Tree; Depth-First Search; Dynamic Programming
class Solution:
    def maximizeSumOfWeights(self, edges: List[List[int]], k: int) -> int:
        def dfs(u: int, fa: int) -> Tuple[int, int]:
            s = 0
            t = []
            for v, w in g[u]:
                if v == fa:
                    continue
                a, b = dfs(v, u)
                s += a
                if (d := (w + b - a)) > 0:
                    t.append(d)
            t.sort(reverse=True)
            return s + sum(t[:k]), s + sum(t[: k - 1])

        n = len(edges) + 1
        g: List[List[Tuple[int, int]]] = [[] for _ in range(n)]
        for u, v, w in edges:
            g[u].append((v, w))
            g[v].append((u, w))
        x, y = dfs(0, -1)
        return max(x, y)
