1. Abridged Problem Statement  
Given a tree of N cities (nodes) and an integer K, select as few “trade centers” (special nodes) as possible so that every city is at distance ≤ K from at least one center. Output the number of centers and their labels.

2. Detailed Editorial  

Overview  
We must cover all nodes in a tree by “balls” of radius K centered at chosen nodes, using the minimum number of centers. A classic greedy strategy on trees is to process from the leaves upward, tracking how far each node is from the nearest center placed so far in its subtree. Whenever a pair of uncovered branches beneath a node would force some leaf-to-leaf path to exceed length 2K without a center, we must place a center there.

Key Idea and DP Definition  
Root the tree at an arbitrary node (say 1). Define dp[u] = the minimum distance from u down to any center placed in u’s subtree; if no center is “close enough,” dp[u] will grow until it triggers placing a center.

Leaf Initialization  
For a leaf u, we pretend there is a center at distance K+1 below it (so dp[u] = K+1). This encodes “no center in its true subtree yet” and allows us to measure when the leaf is too far from any center.

Merging Children  
At an internal node u, collect dp[v] for all children v. Let mn = min(dp[v]) and mx = max(dp[v]). Now:

- If mn + mx + 2 ≤ 2K+1, then the branch closest to a center (distance mn) can cover u and that center still reaches the farthest branch within K after we move up one edge. So we set dp[u] = mn+1.
- Otherwise, the two most distant uncovered tips would form a path longer than 2K if unchecked, so we place a center at u.  Concretely, we detect this when mx+mn+2 > 2K+1; equivalently, when after updating dp[u] = mx+1 we hit dp[u] = 2K+1. At that moment we reset dp[u]=0 (placing a center at u) and record u in our answer.

Final Check at the Root  
After processing the whole tree, if dp[root] > K, it means the root is still too far from any center, so we must add a center at the root.

Proof of Optimality  
This greedy is optimal because placing a center only when forced (a critical path would exceed length 2K without it) ensures minimality. One can show that any solution must have a center in the same positions (up to relabeling) to cover the same critical leaves, so no extra centers are used.

Complexity  
We do a single DFS, merging children in O(deg(u)) each, so O(N) overall. Memory is O(N).

3. C++ Solution

```cpp
#include <bits/stdc++.h>

using namespace std;

template<typename T1, typename T2>
ostream& operator<<(ostream& out, const pair<T1, T2>& x) {
    return out << x.first << ' ' << x.second;
}

template<typename T1, typename T2>
istream& operator>>(istream& in, pair<T1, T2>& x) {
    return in >> x.first >> x.second;
}

template<typename T>
istream& operator>>(istream& in, vector<T>& a) {
    for(auto& x: a) {
        in >> x;
    }
    return in;
};

template<typename T>
ostream& operator<<(ostream& out, const vector<T>& a) {
    for(auto x: a) {
        out << x << ' ';
    }
    return out;
};

int n, k;
vector<vector<int>> adj;
vector<int> dp;

void read() {
    cin >> n >> k;
    adj.assign(n + 1, {});
    dp.assign(n + 1, 0);
    for(int i = 1; i < n; i++) {
        int u, v;
        cin >> u >> v;
        adj[u].push_back(v);
        adj[v].push_back(u);
    }
}

int dfs(int u, int par, vector<int>& ans) {
    vector<int> child_dps;

    for(int v: adj[u]) {
        if(v == par) {
            continue;
        }
        dfs(v, u, ans);
        child_dps.push_back(dp[v]);
    }

    if(!child_dps.empty()) {
        int min_dist = *min_element(child_dps.begin(), child_dps.end());
        int max_dist = *max_element(child_dps.begin(), child_dps.end());

        if(min_dist + max_dist + 2 <= 2 * k + 1) {
            dp[u] = min_dist + 1;
        } else {
            dp[u] = max_dist + 1;
        }
    } else {
        dp[u] = k + 1;
    }

    if(dp[u] == 2 * k + 1) {
        dp[u] = 0;
        ans.push_back(u);
    }

    return dp[u];
}

void solve() {
    // Pick the minimum set of centers so every vertex of the tree is within
    // distance k of some center. Root anywhere (vertex 1): the answer is
    // invariant to the root because rooting at any vertex between two optimal
    // centers gives the same greedy outcome.
    //
    // - Run a post-order DFS where dp[u] is the distance from u up to the
    //   closest center in u's processed subtree. A leaf has no covered subtree,
    //   so it behaves as if a center sits k+1 below it (dp = k+1).
    //
    // - At an internal node take the closest child center distance min_dist and
    //   the farthest still-uncovered one max_dist. If min_dist + max_dist + 2
    //   fits within 2k+1, the closest center also covers the farthest branch,
    //   so dp[u] = min_dist + 1; otherwise the farthest branch is the critical
    //   path and dp[u] = max_dist + 1.
    //
    // - When dp[u] reaches 2k+1 a vertex on the critical path would go
    //   uncovered, so we must place a center at u, resetting dp[u] to 0. After
    //   the DFS, if dp[root] still exceeds k the root subtree is uncovered, so
    //   add the root as a final center.

    vector<int> ans;
    dfs(1, -1, ans);

    if(dp[1] > k) {
        ans.push_back(1);
    }

    sort(ans.begin(), ans.end());

    cout << ans.size() << "\n";
    for(int center: ans) {
        cout << center << "\n";
    }
}

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    int T = 1;
    // cin >> T;
    for(int test = 1; test <= T; test++) {
        read();
        // cout << "Case #" << test << ": ";
        solve();
    }

    return 0;
}
```

4. Python Solution with Detailed Comments

```python
import sys
sys.setrecursionlimit(1000000)

def read_input():
    data = sys.stdin.read().split()
    it = iter(data)
    n, k = int(next(it)), int(next(it))
    adj = [[] for _ in range(n+1)]
    for _ in range(n-1):
        u, v = int(next(it)), int(next(it))
        adj[u].append(v)
        adj[v].append(u)
    return n, k, adj

def solve():
    n, k, adj = read_input()
    dp = [0]*(n+1)
    centers = []

    def dfs(u, p):
        """
        Post-order DFS.
        Computes dp[u]: distance from u to nearest center in subtree.
        Adds to centers[] when a center is placed.
        """
        child_dp = []
        for v in adj[u]:
            if v == p: continue
            dfs(v, u)
            child_dp.append(dp[v])
        if not child_dp:
            # Leaf: no center below → pretend it's at distance k+1
            dp[u] = k + 1
        else:
            mn = min(child_dp)
            mx = max(child_dp)
            # If the two farthest uncovered tips would exceed 2k+1,
            # we must place a center (detected when dp[u] hits 2k+1).
            if mn + mx + 2 <= 2*k + 1:
                dp[u] = mn + 1
            else:
                dp[u] = mx + 1
        # Place center if dp[u] == 2k+1
        if dp[u] == 2*k + 1:
            dp[u] = 0
            centers.append(u)

    dfs(1, 0)
    # After DFS, if root is still > k away, place a center at root
    if dp[1] > k:
        centers.append(1)
    centers.sort()
    # Output
    print(len(centers))
    for c in centers:
        print(c)

if __name__ == "__main__":
    solve()
```

5. Compressed Editorial  
Run one DFS from an arbitrary root. For each node, maintain dp[u] = distance downward to the nearest placed center (initially K+1 at a leaf). When combining children, if the closest and farthest dp values force a leaf-to-leaf path > 2K without coverage, place a center at u (reset dp[u]=0). Finally, if the root remains uncovered beyond K, add it. This greedy is optimal and runs in O(N).