### PROBLEM LINK

### Panel Members

**Problem Setter:** Sergey Nagin

**Problem Tester:** Istvan Nagy

**Editorialist:** Sunny Aggarwal

**Contest Admin:** Praveen Dhinwa

**Russian Translator:** Sergey Nagin

**Mandarin Translator:** Hu Zecong

**Vietnamese Translator:** VNOI Team

**Language Verifier:** Rahul Arora

### DIFFICULTY:

Hard

### PREREQUISITES:

Dynamic Programming, Tree, Memoization, Mathematics.

### PROBLEM:

A tree is an undirected graph on N vertices with N-1 edges and no cycles. Let just consider a peculiar way of comparing two trees. To describe it, let’s start with the way, we have stored a tree. For every tree T, we has a value V — the root of the tree, and for every vertex i, we have an ordered list Q_i with L_i elements — Q_{i, 1}, Q_{i, 2}, ..., Q_{i,L_i} which are children of the vertex i. We assume that the two trees will be equal if their roots are the same and for every i, the ordered list Q_i is the same in both the trees.

Consider the given tree T_1 [V=1, Q_1=[2, 3], Q_2=[], Q_3=[]] and T_2 given as [V=1, Q_1=[3, 2], Q_2=[], Q_3=[]], they will be considered different because Q_1 in the first tree is not equal to Q_1 in the second tree.

Let us consider that the E_i denotes the number of vertices adjacent to vertex i. Given an array C of N elements, Let f(C) be the number of different trees such that there exists a permutation P_1, P_2, ... , P_N so that E_{P_1} = C_1, E_{P_2}= C_2, ... , E_{P_N} = C_N. We are provided with array C and asked to compute the number f(C) modulo 1000000007 (10^9+7).

### EXPLANATION

It is easy to notice that if 1 permutation satisfies a configuration of the tree then every possible permutation will satisfy same configuration. So, we will first compute the number of different trees possible for a given permutation say 1, 2, 3, … N and then will multiply this answer with N!.

Let us consider some notation to understand the solution better. Let us consider a function F(A) denotes the number of different trees possible when there are exactly A_i nodes in each tree with i adjacent nodes. Let us consider another function G(size, A) denotes the number of different tree possible when there are

exactly A_i nodes in each tree with i adjacent nodes but each tree root has exactly size number of adjacent nodes.

**How to calculate G(size, A) ?**

G(size, A) denotes the number of different trees possible (say T) when there are exactly A_i nodes in each tree with i adjacent nodes but each tree root ( say R) has exactly size number of adjacent nodes. We can compute G(size, A) by choosing a subset of A (say Z) and can assign the chosen subset to the first child of root node R and can distribute the remaining set of nodes over the remaining size-1 children of root node.

i.e

**How to calculate F(A) ?**

F(A) is itself dependent on function. F(A) can be calculated as follows:

i.e

where X = updated A with X_i = A_i - 1 as we have selected 1 node with i adjacent nodes as root.

**NOTE:** We have received linear time solutions for this problem during the contest. So, Please have a look on them. This is an overview of author’s approach. I am still not very much clear with the idea so any suggestions and contributions to this post is welcome.

Please have a look at author’s solution to learn more about the discussed solution.

### COMPLEXITY

Exponential Time

### AUTHOR’S AND TESTER’S SOLUTIONS:

The author’s solution can be found here.

The tester’s solution can be found here.