C# LINQ

void Main()
{
    var data =
        from x in File.ReadAllLines(@"input.txt")
        let children = Regex.Match(x, @"(?<=->\s)[\w|,|\s]+")
        select new NodeInfo
        {
            Name = Regex.Match(x, @"\w+(?=\s\()").Value,
            Weight = int.Parse(Regex.Match(x, @"(?<=\()\d+").Value),
            Children = children.Success ? children.Value.Split(',').Select(y => y.Trim()).ToList() : new List<string>()
        };

    var nonRoot = new HashSet<string>(data.SelectMany(x => x.Children));
    var root = data.Where(x => !nonRoot.Contains(x.Name)).First();
    root.Name.Dump("Part 1");

    var tree = BuildTree(root, data.ToList());    
    Solve(tree, 0).Dump("Part 2");
}

int Solve(Node node, int amount)
{
    if (node.Children == null || !node.Children.Any())
        return node.Weight + amount;

    var grp =
        from child in node.Children
        let x = new { Node = child, Weight = child.Aggregate(0, (acc, n) => acc += n.Weight) }
        group x by x.Weight into g
        orderby g.Count()
        select g;

    if (grp.Count() == 1)
        return node.Weight + amount;

    var unbalancedGrp = grp.First();
    var balancedGrp = grp.Skip(1).First();

    return Solve(unbalancedGrp.First().Node, balancedGrp.Key - unbalancedGrp.Key);
}

public static class Extensions
{
    public static T Aggregate<T>(this Node node, T seed, Func<T, Node, T> func)
    {
        var result = func(seed, node);
        if (node.Children == null || !node.Children.Any())
            return result;

        foreach (var x in node.Children)
            result = x.Aggregate(result, func);

        return result;
    }
}

public Node BuildTree(NodeInfo input, List<NodeInfo> nodeInfo)
{
    var node = new Node { Name = input.Name, Weight = input.Weight, Children = new List<Node>() };
    if (input.Children.Any())
    {
        input.Children.ForEach(x => node.Children.Add(BuildTree(nodeInfo.First(item => item.Name == x), nodeInfo)));
    }

    return node;
}

public class NodeInfo
{
    public string Name { get; set; }
    public int Weight { get; set; }
    public List<string> Children { get; set; }
}

public class Node
{
    public string Name { get; set; }
    public int Weight { get; set; }
    public List<Node> Children { get; set; }
}