Problem:
Score : 1000 points

Problem Statement

Snuke has a rooted tree with N+1 vertices.
The vertices are numbered 0 through N, and Vertex 0 is the root of the tree.
The parent of Vertex i (1 \leq i \leq N) is Vertex p_i.

Besides this tree, Snuke also has an box which is initially empty and many marbles, and playing with them.
The play begins with placing one marble on some of the vertices, then proceeds as follows:

If there is a marble on Vertex 0, move the marble into the box.

Move each marble from the vertex to its parent (all at once).

For each vertex occupied by two or more marbles, remove all the marbles from the vertex.

If there exists a vertex with some marbles, go to Step 1. Otherwise, end the play.

There are 2^{N+1} ways to place marbles on some of the vertices.
For each of them, find the number of marbles that will be in the box at the end of the play, and compute the sum of all those numbers modulo 1,000,000,007.

Constraints

1 \leq N < 2 \times 10^{5}

0 \leq p_i < i

Partial Scores

In the test set worth 400 points, N < 2{,}000.

Input

Input is given from Standard Input in the following format:

N
p_1 p_2 ... p_{N}

Output

Print the answer.

Sample Input 1

2
0 0

Sample Output 1

8

When we place a marble on both Vertex 1 and 2, there will be multiple marbles on Vertex 0 by step 2. In such a case, these marbles will be removed instead of being moved to the box.

Sample Input 2

5
0 1 1 0 4

Sample Output 2

96

Sample Input 3

31
0 1 0 2 4 0 4 1 6 4 3 9 7 3 7 2 15 6 12 10 12 16 5 3 20 1 25 20 23 24 23

Sample Output 3

730395550

Be sure to compute the sum modulo 1,000,000,007.

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.io.IOException;
import java.util.InputMismatchException;
import java.io.InputStreamReader;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.Stream;
import java.util.StringTokenizer;
import java.io.BufferedReader;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author amotoma3
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        TaskE solver = new TaskE();
        solver.solve(1, in, out);
        out.close();
    }

    static class TaskE {
        int maxDepth = 0;

        public void solve(int testNumber, InputReader in, PrintWriter out) {
            int n = in.nextInt();
            int[] p = in.nextIntArray(n);
            List<Integer>[] graph = Stream.generate(ArrayList::new).limit(n + 1).toArray(List[]::new);
            for (int i = 0; i < n; i++) {
                graph[p[i]].add(i + 1);
            }

            int[] depth = new int[n + 1];
            rec(graph, 0, 0, depth);

            int[] cnt = new int[maxDepth + 1];
            List<Integer>[] list = Stream.generate(ArrayList::new).limit(maxDepth + 1).toArray(List[]::new);
            for (int i = 0; i <= n; i++) {
                list[depth[i]].add(i);
                cnt[depth[i]]++;
            }

            long MOD = 1_000_000_007;
            Modulo modulo = new Modulo(MOD);
            long ans = 0;
            for (int i = 0; i <= maxDepth; i++) {
                long tmp = modulo.pow(2, n + 1 - cnt[i]);
                long[] dpDeath = new long[n + 1];
                long[] dpOne = new long[n + 1];
                for (int x : list[i]) {
                    dpDeath[x] = 1;
                    dpOne[x] = 1;
                }
                for (int j = i - 1; j >= 0; j--) {
                    for (int x : list[j]) {
                        long deathMlt = 1;
                        for (int child : graph[x]) {
                            deathMlt = modulo.mlt(deathMlt, dpDeath[child]);
                        }
                        long sum = 1;
                        long oneSum = 0;
                        for (int child : graph[x]) {
                            sum = modulo.mlt(sum, modulo.add(dpOne[child], dpDeath[child]));
                            oneSum = modulo.add(oneSum, modulo.mlt(dpOne[child], modulo.mlt(deathMlt, modulo.rev(dpDeath[child]))));
                        }
                        dpOne[x] = oneSum;
                        dpDeath[x] = modulo.sub(sum, oneSum);
                    }
                }
                ans = modulo.add(ans, modulo.mlt(tmp, dpOne[0]));
            }

            out.println(ans);
        }

        void rec(List<Integer>[] graph, int x, int d, int[] depth) {
            depth[x] = d;
            maxDepth = Math.max(maxDepth, d);
            for (int child : graph[x]) {
                rec(graph, child, d + 1, depth);
            }
        }

    }

    static class Modulo {
        long m;

        public Modulo(long m) {
            this.m = m;
        }

        public long add(long a, long b) {
            return (a + b) % m;
        }

        public long sub(long a, long b) {
            return (a - b + m) % m;
        }

        public long mlt(long a, long b) {
            return (a * b) % m;
        }

        public long rev(long a) {
            return pow(a, m - 2);
        }

        public long pow(long a, long x) {
            long ans = 1;
            while (x > 0) {
                if ((x & 1) != 0) {
                    ans = (ans * a) % m;
                }
                x >>= 1;
                a = (a * a) % m;
            }
            return ans;
        }

    }

    static class InputReader {
        BufferedReader in;
        StringTokenizer tok;

        public String nextString() {
            while (!tok.hasMoreTokens()) {
                try {
                    tok = new StringTokenizer(in.readLine(), " ");
                } catch (IOException e) {
                    throw new InputMismatchException();
                }
            }
            return tok.nextToken();
        }

        public int nextInt() {
            return Integer.parseInt(nextString());
        }

        public int[] nextIntArray(int n) {
            int[] res = new int[n];
            for (int i = 0; i < n; i++) {
                res[i] = nextInt();
            }
            return res;
        }

        public InputReader(InputStream inputStream) {
            in = new BufferedReader(new InputStreamReader(inputStream));
            tok = new StringTokenizer("");
        }

    }
}