description: A binary tree of n nodes is given. Nodes of the tree are numbered from 1 to n and the root is the node 1. Each node can have no child, only one left child, only one right child, or both children. For convenience, let's denote l_u and r_u as the left and the right child of the node u respectively, l_u = 0 if u does not have the left child, and r_u = 0 if the node u does not have the right child.
  
  Each node has a string label, initially is a single character c_u. Let's define the string representation of the binary tree as the concatenation of the labels of the nodes in the in-order. Formally, let f(u) be the string representation of the tree rooted at the node u. f(u) is defined as follows: $$$ f(u) = \begin{cases} <empty string>, & if u = 0; \\\ f(l_u) + c_u + f(r_u) & otherwise, \end{cases}  where +$$$ denotes the string concatenation operation.
  
  This way, the string representation of the tree is f(1).
  
  For each node, we can duplicate its label at most once, that is, assign c_u with c_u + c_u, but only if u is the root of the tree, or if its parent also has its label duplicated.
  
  You are given the tree and an integer k. What is the lexicographically smallest string representation of the tree, if we can duplicate labels of at most k nodes?
  
  A string a is lexicographically smaller than a string b if and only if one of the following holds: 
  
    * a is a prefix of b, but a ≠ b; 
    * in the first position where a and b differ, the string a has a letter that appears earlier in the alphabet than the corresponding letter in b. 
  
  Input
  
  The first line contains two integers n and k (1 ≤ k ≤ n ≤ 2 ⋅ 10^5).
  
  The second line contains a string c of n lower-case English letters, where c_i is the initial label of the node i for 1 ≤ i ≤ n. Note that the given string c is not the initial string representation of the tree.
  
  The i-th of the next n lines contains two integers l_i and r_i (0 ≤ l_i, r_i ≤ n). If the node i does not have the left child, l_i = 0, and if the node i does not have the right child, r_i = 0.
  
  It is guaranteed that the given input forms a binary tree, rooted at 1.
  
  Output
  
  Print a single line, containing the lexicographically smallest string representation of the tree if at most k nodes have their labels duplicated.
  
  Examples
  
  Input
  
  
  4 3
  abab
  2 3
  0 0
  0 4
  0 0
  
  
  Output
  
  
  baaaab
  
  Input
  
  
  8 2
  kadracyn
  2 5
  3 4
  0 0
  0 0
  6 8
  0 7
  0 0
  0 0
  
  
  Output
  
  
  daarkkcyan
  
  Input
  
  
  8 3
  kdaracyn
  2 5
  0 3
  0 4
  0 0
  6 8
  0 7
  0 0
  0 0
  
  
  Output
  
  
  darkcyan
  
  Note
  
  The images below present the tree for the examples. The number in each node is the node number, while the subscripted letter is its label. To the right is the string representation of the tree, with each letter having the same color as the corresponding node.
  
  Here is the tree for the first example. Here we duplicated the labels of nodes 1 and 3. We should not duplicate the label of node 2 because it would give us the string "bbaaab", which is lexicographically greater than "baaaab".
  
  <image>
  
  In the second example, we can duplicate the labels of nodes 1 and 2. Note that only duplicating the label of the root will produce a worse result than the initial string.
  
  <image>
  
  In the third example, we should not duplicate any character at all. Even though we would want to duplicate the label of the node 3, by duplicating it we must also duplicate the label of the node 2, which produces a worse result.
  
  <image>
  
  There is no way to produce string "darkkcyan" from a tree with the initial string representation "darkcyan" :(.
solution: import java.io.*;
  import java.util.*;
  
  import static java.lang.Math.*;
  import static java.util.Arrays.*;
  
  public class cf1623e {
  
      public static void main(String[] args) throws IOException {
          int n = rni(), k = ni(), l[] = new int[n], r[] = new int[n], order[] = new int[n], next[] = new int[n], last[] = new int[n];
          fill(next, -1);
          char[] c = rcha();
          boolean good[] = new boolean[n], chosen[] = new boolean[n];
          for (int i = 0; i < n; ++i) {
              l[i] = rni() - 1;
              r[i] = ni() - 1;
          }
          dfs(l, r, 0, order, next, last);
          while (last[lastv] != -1) {
              if (c[lastv] > c[last[lastv]] || good[lastv] && c[lastv] == c[last[lastv]]) {
                  good[last[lastv]] = true;
              }
              lastv = last[lastv];
          }
          dep = new int[n];
          anc = new int[n][20];
          minbad = new int[n][20];
          for (int[] row : anc) {
              fill(row, -1);
          }
          for (int[] row : minbad) {
              fill(row, n);
          }
          binary_lifting_dfs(l, r, 0, good, order);
          int invord[] = new int[n];
          for (int i = 0; i < n; ++i) {
              invord[order[i]] = i;
          }
  //        prln(order);
          for (int x = 0; x < n; ++x) {
              int i = invord[x];
              if (good[i] && dist(i, chosen) <= k && minbad(i, chosen) >= order[i]) {
  //                prln(c[i]);
  //                prln(order[i], dist(i, chosen), minbad(i, chosen));
                  while (i != -1 && !chosen[i]) {
                      chosen[i] = true;
                      --k;
                      i = anc[i][0];
                  }
              }
          }
          StringBuilder ans = new StringBuilder();
          gen(l, r, 0, ans, chosen, c);
          prln(ans);
          close();
      }
  
      static void gen(int l[], int r[], int i, StringBuilder s, boolean chosen[], char[] c) {
          if (l[i] != -1) {
              gen(l, r, l[i], s, chosen, c);
          }
          s.append(c[i]);
          if (chosen[i]) {
              s.append(c[i]);
          }
          if (r[i] != -1) {
              gen(l, r, r[i], s, chosen, c);
          }
      }
  
      static int minbad(int i, boolean chosen[]) {
  //        pr("call ");
  //        prln(i);
          int ans = dep.length;
          for (int j = anc[i].length - 1; j >= 0; --j) {
  //            prln(i, anc[i][j]);
              if (anc[i][j] != -1 && !chosen[anc[i][j]]) {
  //                prln(i, ans, minbad[i][j], minbad[anc[i][j]][0]);
                  ans = min(ans, minbad[i][j]);
                  i = anc[i][j];
              }
          }
          ans = min(ans, minbad[i][0]);
          return ans;
      }
  
      static int dist(int i, boolean chosen[]) {
          int d = dep[i];
          for (int j = anc[i].length - 1; j >= 0; --j) {
              if (anc[i][j] != -1 && !chosen[anc[i][j]]) {
                  i = anc[i][j];
              }
          }
          return d - dep[i] + 1;
      }
  
      // initialization: dep[n], anc[n][lg(n)], fill(anc, -1)
      static int dep[], anc[][], minbad[][];
  
      static void binary_lifting_dfs(int l[], int r[], int i, boolean good[], int order[]) {
          if (!good[i]) {
              minbad[i][0] = order[i];
          }
          for (int j = 1; j < anc[i].length && anc[i][j - 1] >= 0; ++j) {
              anc[i][j] = anc[anc[i][j - 1]][j - 1];
              minbad[i][j] = min(minbad[i][j - 1], minbad[anc[i][j - 1]][j - 1]);
          }
          if (l[i] != -1) {
              dep[l[i]] = dep[i] + 1;
              anc[l[i]][0] = i;
              binary_lifting_dfs(l, r, l[i], good, order);
          }
          if (r[i] != -1) {
              dep[r[i]] = dep[i] + 1;
              anc[r[i]][0] = i;
              binary_lifting_dfs(l, r, r[i], good, order);
          }
      }
  
      static int t = 0, lastv = -1, first = -1;
  
      static void dfs(int l[], int r[], int i, int order[], int next[], int last[]) {
          if (l[i] != -1) {
              dfs(l, r, l[i], order, next, last);
          }
          if (t == 0) {
              first = i;
          }
          order[i] = t++;
          if (lastv != -1) {
              next[lastv] = i;
          }
          last[i] = lastv;
          lastv = i;
          if (r[i] != -1) {
              dfs(l, r, r[i], order, next, last);
          }
      }
  
      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, a}; int[] y = exgcd(b, a % b); return new int[] {y[1], y[0] - y[1] * (a / b), y[2]};}
      static long[] exgcd(long a, long b) {if (b == 0) return new long[] {1, 0, a}; long[] y = exgcd(b, a % b); return new long[] {y[1], y[0] - y[1] * (a / b), y[2]};}
      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();}
  }