# S-Trees

Description

A Strange Tree (S-tree) over the variable set \(X_n = \{x_1,x_2,\dots,x_n\}\) is a binary tree representing a Boolean function \(f : \{0,1\}^n\to\{0,1\}\). Each path of the S-tree begins at the root node and consists of n+1 nodes. Each of the S-tree's nodes has a depth, which is the amount of nodes between itself and the root (so the root has depth 0). The nodes with depth less than n are called non-terminal nodes. All non-terminal nodes have two children: the right child and the left child. Each non-terminal node is marked with some variable xi from the variable set X_{n}. All non-terminal nodes with the same depth are marked with the same variable, and non-terminal nodes with different depth are marked with different variables. So, there is a unique variable x_{i1} corresponding to the root, a unique variable x_{i2} corresponding to the nodes with depth 1, and so on. The sequence of the variables \(x_{i1},x_{i2},\dots,x_{in}\) is called the variable ordering. The nodes having depth n are called terminal nodes. They have no children and are marked with either 0 or 1. Note that the variable ordering and the distribution of 0's and 1's on terminal nodes are sufficient to completely describe an S-tree.

As stated earlier, each S-tree represents a Boolean function f. If you have an S-tree and values for the variables \(x_{1},x_{2},\dots,x_{n}\), then it is quite simple to find out what \(f(x_{1},x_{2},\dots,x_{n})\) is: start with the root. Now repeat the following: if the node you are at is labelled with a variable xi, then depending on whether the value of the variable is 1 or 0, you go its right or left child, respectively. Once you reach a terminal node, its label gives the value of the function.

Figure 1: S-trees for the x_{1} and (x_{2} or x_{3}) function

On the picture, two S-trees representing the same Boolean function, \(f(x_{1},x_{2},x_{3}) = x_1\) and (x_{2} or x_{3}), are shown. For the left tree, the variable ordering is \(x_{1},x_{2},x_{3}\), and for the right tree it is \(x_{3},x_{1},x_{2}\).

The values of the variables \(x_{1},x_{2},\dots,x_{n}\), are given as a *Variable Values Assignment* (VVA)

\((x_1=b_1,x_2=b_2,\dots,,x_n=b_n)\)

with \(b_{1},b_{2},\dots,b_{n}\) in {0,1}. For instance, ( x_{1} = 1, x_{2} = 1, x_{3} = 0) would be a valid VVA for n = 3, resulting for the sample function above in the value \(f(1,1,0) = 1\) and (1 or 0) = 1. The corresponding paths are shown bold in the picture.

Your task is to write a program which takes an S-tree and some VVAs and computes \(f(x_{1},x_{2},\dots,x_{n})\) as described above.

Input

The input contains the description of several S-trees with associated VVAs which you have to process. Each description begins with a line containing a single integer n, 1 <= n <= 7, the depth of the S-tree. This is followed by a line describing the variable ordering of the S-tree. The format of that line is \(x_{i1},x_{i2},\dots,x_{in}\)(There will be exactly n different space-separated strings). So, for n = 3 and the variable ordering \(x_{3},x_{1},x_{2}\), this line would look as follows:

\(x_{3},x_{1},x_{2}\)

In the next line the distribution of 0's and 1's over the terminal nodes is given. There will be exactly 2^n characters (each of which can be 0 or 1), followed by the new-line character. The characters are given in the order in which they appear in the S-tree, the first character corresponds to the leftmost terminal node of the S-tree, the last one to its rightmost terminal node.

The next line contains a single integer m, the number of VVAs, followed by m lines describing them. Each of the m lines contains exactly n characters (each of which can be 0 or 1), followed by a new-line character. Regardless of the variable ordering of the S-tree, the first character always describes the value of x1, the second character describes the value of x2, and so on. So, the line

110

corresponds to the VVA ( x_{1} = 1, x_{2} = 1, x_{3} = 0).

The input is terminated by a test case starting with n = 0. This test case should not be processed.

Output

For each S-tree, output the line "S-Tree #j:", where j is the number of the S-tree. Then print a line that contains the value of \(f(x_{1},x_{2},\dots,x_{n})\)for each of the given m VVAs, where f is the function defined by the S-tree.

Output a blank line after each test case.

Sample Input

3 x1 x2 x3 00000111 4 000 010 111 110 3 x3 x1 x2 00010011 4 000 010 111 110 0

Sample Output

S-Tree #1: 0011 S-Tree #2: 0011

Source

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