[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data Dependence
   All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.

## 4. Example Code Snippet

### Example 1

```python
1    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Python
1   from collections import deque
2   MOD = 10 ** 9 + 7
3   def solve(n, edge_list):
4       factorial = [1] * n
5       for i in range(1, n):
6           factorial[i] = (factorial[i - 1] * i) % MOD
7       factorial_inv = [1] * n
8       factorial_inv[-1] = pow(factorial[-1], MOD - 2, MOD)
9       for i in range(n - 2, 0, -1):
10          factorial_inv[i] = (factorial_inv[i + 1] * (i + 1)) % MOD
11      g = [[] for _ in range(n)]
12      for i in range(n - 1):
13          a, b = edge_list[i]
14          g[a].append(b)
15          g[b].append(a)
16      order = []
17      parent = [-1] * n
18      parent[0] = 0
19      children = [[] for _ in range(n)]
20      queue = deque([0])
21      while len(queue) > 0:
22          p = queue.popleft()
23          order.append(p)
24          for q in g[p]:
25              if parent[q] == -1:
26                  parent[q] = p
27                  children[p].append(q)
28                  queue.append(q)
29      count_children_res = [-1] * n
30      for x in order[::-1]:
31          if len(children[x]) == 0:
32              count_children_res[x] = 0
33          else:
34              total = 0
35              for y in children[x]:
36                  total += count_children_res[y]
37              r = len(children[x]) + total
38              count_children_res[x] = r
39      assign_children_res = [-1] * n
40      for x in order[::-1]:
41          if len(children[x]) == 0:
42              assign_children_res[x] = 1
43          else:
44              r = factorial[count_children_res[x]]
45              for y in children[x]:
46                  r *= assign_children_res[y]
47                  r *= factorial_inv[count_children_res[y] + 1]
48                  r %= MOD
49              assign_children_res[x] = r
50      assign_parents_res = [-1] * n
51      assign_parents_res[0] = 1
52      for x in order[1:]:
53          p = parent[x]
54          r = assign_children_res[p]
55          r *= pow(assign_children_res[x], MOD - 2, MOD)
56          r *= factorial[count_children_res[x] + 1]
57          r *= factorial_inv[count_children_res[p]]
58          r *= assign_parents_res[p]
59          r *= factorial[n - count_children_res[x] - 2]
60          r *= factorial_inv[n - count_children_res[p] - 1]
61          r %= MOD
62          assign_parents_res[x] = r
63      res_list = [0] * n
64      res_list[0] = assign_children_res[0]
65      for i in range(1, n):
66          res = factorial[n - 1]
67          for j in children[i]:
68              res *= assign_children_res[j]
69              res *= factorial_inv[count_children_res[j] + 1]
70              res %= MOD
71          res *= assign_parents_res[i]
72          res *= factorial_inv[n - 1 - count_children_res[i]]
73          res %= MOD
74          res_list[i] = res
75      return res_list
76  def main():
77      n = int(input())
78      edge_list = []
79      for _ in range(n - 1):
80          a, b = map(int, input().split())
81          edge_list.append((a - 1, b - 1))
82      res_list = solve(n, edge_list)
83      for res in res_list:
84          print(res)
85  def test():
86      assert solve(3, [(0, 1), (0, 2)]) == [2, 1, 1]
87      assert solve(2, [(0, 1)]) == [1, 1]
88      assert solve(5, [(0, 1), (1, 2), (2, 3), (2, 4)]) == [2, 8, 12, 3, 3]
89      assert solve(8, [(0, 1), (1, 2), (2, 3), (2, 4), (2, 5), (5, 6), (5, 7)]) == [40, 280, 840, 120, 120, 504, 72, 72]
90  if __name__ == "__main__":
91      test()
92      main()
```


**Question**: Does `(x,52)` have data dependence over `(p,53)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes