description: Two players, Red and Blue, are at it again, and this time they're playing with crayons! The mischievous duo is now vandalizing a rooted tree, by coloring the nodes while playing their favorite game.
  
  The game works as follows: there is a tree of size n, rooted at node 1, where each node is initially white. Red and Blue get one turn each. Red goes first. 
  
  In Red's turn, he can do the following operation any number of times: 
  
    * Pick any subtree of the rooted tree, and color every node in the subtree red. 
  
  However, to make the game fair, Red is only allowed to color k nodes of the tree. In other words, after Red's turn, at most k of the nodes can be colored red.
  
  Then, it's Blue's turn. Blue can do the following operation any number of times: 
  
    * Pick any subtree of the rooted tree, and color every node in the subtree blue. However, he's not allowed to choose a subtree that contains a node already colored red, as that would make the node purple and no one likes purple crayon. 
  
  Note: there's no restriction on the number of nodes Blue can color, as long as he doesn't color a node that Red has already colored.
  
  After the two turns, the score of the game is determined as follows: let w be the number of white nodes, r be the number of red nodes, and b be the number of blue nodes. The score of the game is w ⋅ (r - b).
  
  Red wants to maximize this score, and Blue wants to minimize it. If both players play optimally, what will the final score of the game be?
  
  Input
  
  The first line contains two integers n and k (2 ≤ n ≤ 2 ⋅ 10^5; 1 ≤ k ≤ n) — the number of vertices in the tree and the maximum number of red nodes.
  
  Next n - 1 lines contains description of edges. The i-th line contains two space separated integers u_i and v_i (1 ≤ u_i, v_i ≤ n; u_i ≠ v_i) — the i-th edge of the tree.
  
  It's guaranteed that given edges form a tree.
  
  Output
  
  Print one integer — the resulting score if both Red and Blue play optimally.
  
  Examples
  
  Input
  
  
  4 2
  1 2
  1 3
  1 4
  
  
  Output
  
  
  1
  
  
  Input
  
  
  5 2
  1 2
  2 3
  3 4
  4 5
  
  
  Output
  
  
  6
  
  
  Input
  
  
  7 2
  1 2
  1 3
  4 2
  3 5
  6 3
  6 7
  
  
  Output
  
  
  4
  
  
  Input
  
  
  4 1
  1 2
  1 3
  1 4
  
  
  Output
  
  
  -1
  
  Note
  
  In the first test case, the optimal strategy is as follows: 
  
    * Red chooses to color the subtrees of nodes 2 and 3. 
    * Blue chooses to color the subtree of node 4. 
  
  At the end of this process, nodes 2 and 3 are red, node 4 is blue, and node 1 is white. The score of the game is 1 ⋅ (2 - 1) = 1.
  
  In the second test case, the optimal strategy is as follows: 
  
    * Red chooses to color the subtree of node 4. This colors both nodes 4 and 5. 
    * Blue does not have any options, so nothing is colored blue. 
  
  At the end of this process, nodes 4 and 5 are red, and nodes 1, 2 and 3 are white. The score of the game is 3 ⋅ (2 - 0) = 6.
  
  For the third test case:
  
  <image>
  
  The score of the game is 4 ⋅ (2 - 1) = 4.
solution: import java.io.*;
  import java.util.*;
  
  import static java.lang.Math.*;
  import static java.util.Arrays.*;
  
  public class cf1615e_2 {
  
      public static void main(String[] args) throws IOException {
          int n = rni(), k = ni(), dep[] = new int[n], par[] = new int[n], mxd[] = new int[n], goodcnt = n, taken = 0;
          boolean leaf[] = new boolean[n], bad[] = new boolean[n];
          Graph g = tree(n);
          par[0] = -1;
          dfs(g, 0, par, dep, mxd, leaf);
          PriorityQueue<int[]> pq = new PriorityQueue<>((a, b) -> b[1] - a[1]);
          pq.offer(new int[] {0, mxd[0]});
          while (!pq.isEmpty() && taken < k && goodcnt > 0) {
              int e[] = pq.poll(), i = e[0];
              ++taken;
              while (i != -1) {
                  bad[i] = true;
                  --goodcnt;
                  int nxt = -1;
                  for (int j : g.get(i)) {
                      if (j != par[i]) {
                          if (mxd[j] == mxd[i] - 1 && nxt == -1) {
                              nxt = j;
                          } else {
                              pq.offer(new int[] {j, mxd[j]});
                          }
                      }
                  }
                  i = nxt;
              }
          }
          // prln(n, goodcnt, taken);
          if (goodcnt == 0) {
              long ans = (long) taken * (n - taken);
              for (long i = taken + 1; i <= k; ++i) {
                  if (i * (n - i) > ans) {
                      ans = i * (n - i);
                  }
              }
              prln(ans);
              close();
              return;
          }
          if (goodcnt > taken) {
              long ans = (long) (n - goodcnt - taken) * (taken - goodcnt);
              for (long i = goodcnt - 1; i >= 0; --i) {
                  long pans = (n - i - taken) * (taken - i);
                  if (pans < ans) {
                      ans = pans;
                  }
              }
              prln(ans);
          } else {
              prln((long) (n - goodcnt - taken) * (taken - goodcnt));
          }
          close();
      }
  
      static void dfs(Graph g, int i, int par[], int dep[], int mxd[], boolean leaf[]) {
          if (g.get(i).size() == 1) {
              mxd[i] = 0;
              leaf[i] = true;
          }
          for (int j : g.get(i)) {
              if (j != par[i]) {
                  dep[j] = dep[i] + 1;
                  par[j] = i;
                  dfs(g, j, par, dep, mxd, leaf);
                  mxd[i] = max(mxd[i], 1 + mxd[j]);
              }
          }
      }
  
      static Graph graph(int n) {
          Graph g = new Graph();
          for (int i = 0; i < n; ++i) {
              g.add(new ArrayList<>());
          }
          return g;
      }
  
      static Graph graph(int n, int m) throws IOException {
          Graph g = graph(n);
          for (int i = 0; i < m; ++i) {
              g.c(rni() - 1, ni() - 1);
          }
          return g;
      }
  
      static Graph digraph(int n, int m) throws IOException {
          Graph g = graph(n);
          for (int i = 0; i < m; ++i) {
              g.cto(rni() - 1, ni() - 1);
          }
          return g;
      }
  
      static Graph tree(int n) throws IOException {
          return graph(n, n - 1);
      }
  
      static Graph graph(List<? extends Collection<Integer>> g) {
          int n = g.size();
          Graph h = graph(n);
          for (int i = 0; i < n; ++i) {
              for (int j : g.get(i)) {
                  h.cto(i, j);
              }
          }
          return h;
      }
  
      static class Graph extends ArrayList<List<Integer>> {
          void cto(int u, int v) {
              get(u).add(v);
          }
  
          void c(int u, int v) {
              cto(u, v);
              cto(v, u);
          }
      }
  
      static BufferedReader __i = new BufferedReader(new InputStreamReader(System.in));
      static PrintWriter __o = new PrintWriter(new OutputStreamWriter(System.out));
      static StringTokenizer input;
      static Random __r = new Random();
  
      // references
      // IBIG = 1e9 + 7
      // IMAX ~= 2e9
      // LMAX ~= 9e18
  
      // constants
      static final int IBIG = 1000000007;
      static final int IMAX = 2147483647;
      static final long LMAX = 9223372036854775807L;
      // math util
      static int minof(int a, int b, int c) {return min(a, min(b, c));}
      static int minof(int... x) {if (x.length == 1) return x[0]; if (x.length == 2) return min(x[0], x[1]); if (x.length == 3) return min(x[0], min(x[1], x[2])); int min = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] < min) min = x[i]; return min;}
      static long minof(long a, long b, long c) {return min(a, min(b, c));}
      static long minof(long... x) {if (x.length == 1) return x[0]; if (x.length == 2) return min(x[0], x[1]); if (x.length == 3) return min(x[0], min(x[1], x[2])); long min = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] < min) min = x[i]; return min;}
      static int maxof(int a, int b, int c) {return max(a, max(b, c));}
      static int maxof(int... x) {if (x.length == 1) return x[0]; if (x.length == 2) return max(x[0], x[1]); if (x.length == 3) return max(x[0], max(x[1], x[2])); int max = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] > max) max = x[i]; return max;}
      static long maxof(long a, long b, long c) {return max(a, max(b, c));}
      static long maxof(long... x) {if (x.length == 1) return x[0]; if (x.length == 2) return max(x[0], x[1]); if (x.length == 3) return max(x[0], max(x[1], x[2])); long max = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] > max) max = x[i]; return max;}
      static int powi(int a, int b) {if (a == 0) return 0; int ans = 1; while (b > 0) {if ((b & 1) > 0) ans *= a; a *= a; b >>= 1;} return ans;}
      static long powl(long a, int b) {if (a == 0) return 0; long ans = 1; while (b > 0) {if ((b & 1) > 0) ans *= a; a *= a; b >>= 1;} return ans;}
      static int fli(double d) {return (int) d;}
      static int cei(double d) {return (int) ceil(d);}
      static long fll(double d) {return (long) d;}
      static long cel(double d) {return (long) ceil(d);}
      static int gcd(int a, int b) {return b == 0 ? a : gcd(b, a % b);}
      static long gcd(long a, long b) {return b == 0 ? a : gcd(b, a % b);}
      static int[] exgcd(int a, int b) {if (b == 0) return new int[] {1, 0}; int[] y = exgcd(b, a % b); return new int[] {y[1], y[0] - y[1] * (a / b)};}
      static long[] exgcd(long a, long b) {if (b == 0) return new long[] {1, 0}; long[] y = exgcd(b, a % b); return new long[] {y[1], y[0] - y[1] * (a / b)};}
      static int randInt(int min, int max) {return __r.nextInt(max - min + 1) + min;}
      static long mix(long x) {x += 0x9e3779b97f4a7c15L; x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9L; x = (x ^ (x >> 27)) * 0x94d049bb133111ebL; return x ^ (x >> 31);}
      // array util
      static void reverse(int[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {int swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
      static void reverse(long[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {long swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
      static void reverse(double[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {double swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
      static void reverse(char[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {char swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
      static void shuffle(int[] a) {int n = a.length - 1; for (int i = 0; i < n; ++i) {int ind = randInt(i, n); int swap = a[i]; a[i] = a[ind]; a[ind] = swap;}}
      static void shuffle(long[] a) {int n = a.length - 1; for (int i = 0; i < n; ++i) {int ind = randInt(i, n); long swap = a[i]; a[i] = a[ind]; a[ind] = swap;}}
      static void shuffle(double[] a) {int n = a.length - 1; for (int i = 0; i < n; ++i) {int ind = randInt(i, n); double swap = a[i]; a[i] = a[ind]; a[ind] = swap;}}
      static void rsort(int[] a) {shuffle(a); sort(a);}
      static void rsort(long[] a) {shuffle(a); sort(a);}
      static void rsort(double[] a) {shuffle(a); sort(a);}
      static int[] copy(int[] a) {int[] ans = new int[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
      static long[] copy(long[] a) {long[] ans = new long[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
      static double[] copy(double[] a) {double[] ans = new double[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
      static char[] copy(char[] a) {char[] ans = new char[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
      // input
      static void r() throws IOException {input = new StringTokenizer(rline());}
      static int ri() throws IOException {return Integer.parseInt(rline());}
      static long rl() throws IOException {return Long.parseLong(rline());}
      static double rd() throws IOException {return Double.parseDouble(rline());}
      static int[] ria(int n) throws IOException {int[] a = new int[n]; r(); for (int i = 0; i < n; ++i) a[i] = ni(); return a;}
      static void ria(int[] a) throws IOException {int n = a.length; r(); for (int i = 0; i < n; ++i) a[i] = ni();}
      static int[] riam1(int n) throws IOException {int[] a = new int[n]; r(); for (int i = 0; i < n; ++i) a[i] = ni() - 1; return a;}
      static void riam1(int[] a) throws IOException {int n = a.length; r(); for (int i = 0; i < n; ++i) a[i] = ni() - 1;}
      static long[] rla(int n) throws IOException {long[] a = new long[n]; r(); for (int i = 0; i < n; ++i) a[i] = nl(); return a;}
      static void rla(long[] a) throws IOException {int n = a.length; r(); for (int i = 0; i < n; ++i) a[i] = nl();}
      static double[] rda(int n) throws IOException {double[] a = new double[n]; r(); for (int i = 0; i < n; ++i) a[i] = nd(); return a;}
      static void rda(double[] a) throws IOException {int n = a.length; r(); for (int i = 0; i < n; ++i) a[i] = nd();}
      static char[] rcha() throws IOException {return rline().toCharArray();}
      static void rcha(char[] a) throws IOException {int n = a.length, i = 0; for (char c : rline().toCharArray()) a[i++] = c;}
      static String rline() throws IOException {return __i.readLine();}
      static String n() {return input.nextToken();}
      static int rni() throws IOException {r(); return ni();}
      static int ni() {return Integer.parseInt(n());}
      static long rnl() throws IOException {r(); return nl();}
      static long nl() {return Long.parseLong(n());}
      static double rnd() throws IOException {r(); return nd();}
      static double nd() {return Double.parseDouble(n());}
      // output
      static void pr(int i) {__o.print(i);}
      static void prln(int i) {__o.println(i);}
      static void pr(long l) {__o.print(l);}
      static void prln(long l) {__o.println(l);}
      static void pr(double d) {__o.print(d);}
      static void prln(double d) {__o.println(d);}
      static void pr(char c) {__o.print(c);}
      static void prln(char c) {__o.println(c);}
      static void pr(char[] s) {__o.print(new String(s));}
      static void prln(char[] s) {__o.println(new String(s));}
      static void pr(String s) {__o.print(s);}
      static void prln(String s) {__o.println(s);}
      static void pr(Object o) {__o.print(o);}
      static void prln(Object o) {__o.println(o);}
      static void prln() {__o.println();}
      static void pryes() {prln("yes");}
      static void pry() {prln("Yes");}
      static void prY() {prln("YES");}
      static void prno() {prln("no");}
      static void prn() {prln("No");}
      static void prN() {prln("NO");}
      static boolean pryesno(boolean b) {prln(b ? "yes" : "no"); return b;};
      static boolean pryn(boolean b) {prln(b ? "Yes" : "No"); return b;}
      static boolean prYN(boolean b) {prln(b ? "YES" : "NO"); return b;}
      static void prln(int... a) {for (int i = 0, len = a.length - 1; i < len; pr(a[i]), pr(' '), ++i); if (a.length > 0) prln(a[a.length - 1]); else prln();}
      static void prln(long... a) {for (int i = 0, len = a.length - 1; i < len; pr(a[i]), pr(' '), ++i); if (a.length > 0) prln(a[a.length - 1]); else prln();}
      static void prln(double... a) {for (int i = 0, len = a.length - 1; i < len; pr(a[i]), pr(' '), ++i); if (a.length > 0) prln(a[a.length - 1]); else prln();}
      static <T> void prln(Collection<T> c) {int n = c.size() - 1; Iterator<T> iter = c.iterator(); for (int i = 0; i < n; pr(iter.next()), pr(' '), ++i); if (n >= 0) prln(iter.next()); else prln();}
      static void h() {prln("hlfd"); flush();}
      static void flush() {__o.flush();}
      static void close() {__o.close();}
  }