Is it a Tree?

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Points: 7 (partial)
Time limit: 1.0s
Memory limit: 256M
Authors:
Xyene, FatalEagle
Problem types

Mine is playing a game with Tatsumi, where one person sketches some branches and knots and the other determines whether the sketch is of a tree or not. After playing this for a few hours, they've gotten bored and have decided to make the game more abstract. Now, instead of drawing pictures of trees, the couple fills in a 4×4 adjacency matrix with either 0 or 1.

Their adjacency matrix stores the relationship between any two knots u and v. Specifically, if adj[u][v]=1, then there is a branch connecting u→v. Since branches are undirected, a branch from u→v implies a branch from v→u. In this way, the adjacency matrix is symmetrical in that adj[u][v]=adj[v][u].

An adjacency matrix representing a tree has exactly one path between any pair of u and v, and hence has 1 fewer branches than it has knots. Any less and the tree would be disconnected (a forest), and any more and a cycle would exist.

You've decided to join in the fun. Given a 4×4 adjacency matrix, can you determine whether it represents a tree or not?

Input Specification

An adjacency matrix, represented by 4 lines, each containing 4 space-separated integers, either 0 or 1.

Output Specification

Yes if the given adjacency matrix represents a tree, or No otherwise.

Tips

If you are unfamiliar with the definition of a tree, this Wikipedia article might help.

Sample Input 1

Copy
0 0 0 1
0 0 0 1
0 0 0 1
1 1 1 0

Sample Output 1

Copy
Yes

Sample Input 2

Copy
0 1 0 1
1 0 0 1
0 0 0 1
1 1 1 0

Sample Output 2

Copy
No

Explanation of Output for Sample Input 2

Between Sample 1 and Sample 2, there is a difference of one branch: 0⟷1. This branch creates a cycle, and therefore the number of knots is the same as the number of branches. Hence, it is not a tree.

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