Giving Tree of Errors: Validate Parent-Child Pairs and Output Lexicographically Smallest S-Expression

You are given an unordered list of parent-child pairs describing edges of a binary tree. Each edge is formatted as (P,C) meaning P is the parent of C. Node names are uppercase letters A-Z. Tasks: Determine whether the edges form a valid binary tree. If invalid, output exactly one error code E1~E5 according to the given priority. If valid, build the tree and output the lexicographically smallest S-Expression. For each node, output (X<left><right>), ordering children alphabetically; if a node has only one child, it is placed in the left position. Validity rules (common OA variant) A parent may have at most two children. Duplicate edges are not allowed. A child may have at most one parent. There must be exactly one root (exactly one node with in-degree 0) and the structure must be connected as a tree (not a forest). No cycles. Input (suggested) One line containing pairs separated by spaces, e.g.: (A,B) (A,C) (B,D) Output If invalid: print E1/E2/E3/E4/E5. If valid: print the lexicographically smallest S-Expression. Examples Input: (A,B) (A,C) (B,D) Output: (A(B(D))(C)) Input: (A,B) (A,B) Output: E2 Input: (A,B) (C,D) Output: E4 Example Input (A,B) (A,C) (B,D) Output (A(B(D))(C))